UG Applied Mathematics Booster Test 2 - Fundamentals of Integration -

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UG Applied Mathematics Booster Test 2 - Fundamentals of Integration

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Applied Mathematics Booster Test 2 - Fundamentals of Integration

1 / 20

QUESTION 1 OF 20

By identifying the corresponding primitive functions, compute the integral:
\(\int \left(x,\ \frac{1}{x}\right) dx.\)

Answer Review

1 By identifying the corresponding primitive functions, compute the integral:
\(\int \left(x,\ \frac{1}{x}\right) dx.\)

A \(\frac{x^{2}}{2}-\frac{1}{x^{2}}+C\)

B \(\frac{x^{2}}{2}+log⁡∣x∣+C\)

C \(1+log⁡∣x∣+C\)

D \(\frac{x^{2}}{2}+e^{x}+C\)

💡 Short Explanation

Integrate each term separately. ∫x dx = x²/2 and ∫1/x dx = log|x|. Add the constant of integration C.

✅ Correct Answer Explanation

The integral follows the linearity property of integration. Integrating x gives x²/2, while integrating 1/x gives log|x|. Therefore, the required primitive function is x²/2 + log|x| + C. Option B correctly includes both terms and the constant of integration. Option A incorrectly integrates 1/x, C omits the integral of x, and D incorrectly replaces 1/x with an exponential function.

❓ Why Other Options Are Incorrect

Option A) x² / 2 - 1/x² + C → The integral of 1/x is log|x|, not −1/x². The result incorrectly applies the power rule to a function where it is not applicable.
Option C) 1 + log|x| + C → This omits the integral of x, which should be x²/2. Therefore, the antiderivative is incomplete.
Option D) x² / 2 + e^x + C → The exponential term has no relation to integrating 1/x. Since the derivative of e^x is e^x, this answer is mathematically incorrect.

🎯 Strategy

Used: Elimination

Application: First identify the standard integrals ∫x dx and ∫1/x dx. Eliminate options containing incorrect antiderivatives or missing terms.

Final Logic: Only Option B correctly combines both standard integrals and includes the arbitrary constant.

🧠 Memory Trick

"Power → x²/2, Reciprocal → log|x|."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Indefinite Integrals and Standard Integrals

2 Match the specific indefinite integral in List I with its correct evaluated form in List II.

List I	List II
1. \(\int x^{3} dx\)	I. \(\frac{1}{2a}log⁡∣\frac{x-a}{x+a}∣+C\)
2. \(\int \frac{1}{x^{2}-a^{2}} dx\)	II. \(\frac{x^{2}}{2}+C\)
3. \(\int e^{x} dx\)	III. \(\frac{x^{4}}{4}+C\)
4. \(\int x dx\)	IV. \(e^{x}+C\)

A (A)-(III), (B)-(I), (C)-(IV), (D)-(II)

B (A)-(I), (B)-(II), (C)-(III), (D)-(IV)

C (A)-(II), (B)-(III), (C)-(I), (D)-(IV)

D (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

💡 Short Explanation

Using standard integral formulas: \(\int x^{3}dx=\frac{x^{4}}{4}+C\) \(\int \frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}log⁡∣\frac{x-a}{x+a}∣+C\) \(\int e^{x}dx=e^{x}+C\) \(\int xdx=\frac{x^{2}}{2}+C\) Thus, the correct matching is A–III, B–I, C–IV, D–II.

✅ Correct Answer Explanation

Match each integral with its standard result: (A) \(\int x^{3} dx=\frac{x^{4}}{4}+C\) → (III) (B) \(\int \frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}log⁡∣\frac{x-a}{x+a}∣+C\) → (I) (C) \(\int e^{x} dx=e^{x}+C\) → (IV) (D) \(\int x dx=\frac{x^{2}}{2}+C\) → (II) Hence, the correct matching is \((A)-(III), (B)-(I), (C)-(IV), (D)-(II).\) Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option A)
✅ Correct.
Every integral is matched with its correct standard formula.
Option B)
Incorrect because:
\(\int x^{3}dx\) does not equal the logarithmic expression.
\(\int e^{x}dx\) does not equal \(\frac{x^{4}}{4}+C\).
Option C)
Incorrect because:
\(\int x^{3}dx\) is not \(\frac{x^{2}}{2}+C\).
\(\int \frac{1}{x^{2}-a^{2}}dx\) is not \(\frac{x^{4}}{4}+C\).
Option D)
Incorrect because:
\(\int x^{3}dx\) is not \(e^{x}+C\).
\(\int xdx\) is not \(\frac{x^{4}}{4}+C\).

🎯 Strategy

Used

Recall Standard Integral Formulae

Application:

1. Use the power rule for polynomial functions.
2. Remember the exponential integral:
\(\int e^{x}dx=e^{x}+C.\)
1. Recall the logarithmic formula for
\(\int \frac{1}{x^{2}-a^{2}}dx.\)
1. Match each result accordingly.

🧠 Memory Trick

Mnemonic: "Power Grows, Difference Logs, Exponential Stays."

📚 Subtopic: Standard Integral Formulae – Polynomial, Exponential and Logarithmic Integrals

3 When using the method of substitution to transform an integral ∫ f(x) dx by substituting x = g(t), which of the following operations correctly apply?
1. The independent variable x is changed to t.
2. The differential term dx is replaced by g'(t) dt.
3. The new transformed integral becomes ∫ f(g(t)) g'(t) dt.
4. The arbitrary constant of integration C is no longer required.

A 1, 2, 4

B 1, 2

C 1, 2, 3

D 1, 2, 3, 4

💡 Short Explanation

Substitution changes the variable of integration. Replace dx using the derivative of the substitution. The constant C is still required in an indefinite integral.

✅ Correct Answer Explanation

In substitution, the original variable x is replaced by a new variable t, and the differential changes according to dx = g'(t)dt. Consequently, the integral transforms into ∫ f(g(t))g'(t)dt. However, the arbitrary constant C remains necessary because indefinite integrals represent a family of antiderivatives. Therefore, statements 1, 2, and 3 are correct, while 4 is incorrect, making Option C the correct answer.

❓ Why Other Options Are Incorrect

Option A) 1, 2, 4 → Although statements 1 and 2 are correct, statement 4 is false because the constant of integration must always be included in an indefinite integral.
Option B) 1, 2 → This option omits statement 3, which correctly represents the transformed integral after substitution. Hence, it is incomplete.
Option D) 1, 2, 3, 4 → Statement 4 is incorrect since substitution does not eliminate the arbitrary constant. Therefore, this option contains an incorrect statement.

🎯 Strategy

Used: Elimination

Application: Verify each statement individually using the substitution method. Eliminate every option containing the incorrect statement that removes the constant of integration.

Final Logic: Since only statements 1, 2, and 3 are valid, Option C is the only correct choice.

🧠 Memory Trick

"Substitute – Replace – Transform – Keep C."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Method of Substitution for Indefinite Integrals

4 Which of the following statements is INCORRECT regarding the properties and notation of indefinite integrals?
\(\frac{d}{dx}\left[\int f(x) dx\right]\)
evaluates to \(f(x)+C\)

A The expression f(x) dx is called the element of integration.

B The term C is required because multiple functions can share the exact same derivative.

C The operation

D The integral sign is visually represented by the symbol ∫.

💡 Short Explanation

Differentiation and integration are inverse operations. The derivative of a constant is zero. Therefore, the derivative of an indefinite integral is simply f(x).

✅ Correct Answer Explanation

If ∫f(x)dx = F(x) + C, then differentiating gives d/dx[F(x)+C] = F'(x) = f(x) because the derivative of the constant C is zero. Hence, Option C is the incorrect statement. Options A, B, and D correctly describe the notation and properties of indefinite integrals according to NCERT.

❓ Why Other Options Are Incorrect

Option A) The expression f(x) dx is called the element of integration. → This is a correct statement. The expression f(x)dx represents the integrand together with the differential element used during integration.
Option B) The term C is required because multiple functions can share the exact same derivative. → This is correct because adding any constant to an antiderivative does not change its derivative, necessitating the arbitrary constant.
Option D) The integral sign is visually represented by the symbol ∫. → This is the standard mathematical notation used universally for integration and is correctly stated.

🎯 Strategy

Used: Extreme Word Filter

Application: Check whether each mathematical statement is universally true. The expression f(x) + C after differentiation violates the basic derivative rule for constants.

Final Logic: Since the derivative of C is zero, Option C is the only incorrect statement.

🧠 Memory Trick

"Differentiate ⇒ Constant disappears."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Indefinite Integrals and Properties of Antiderivatives

5 Considering the derivative curve f(x) = 2x, its anti-derivative is y = x² + C. If a specific graph displays the curve y = x² − 1, which arbitrary constant C characterizes this exact member of the curve family?

A C = 0

B C = 1

C C = -1

D C = 2

💡 Short Explanation

The general antiderivative is y = x² + C. Compare it with the given equation y = x² − 1. Therefore, the constant of integration is −1.

✅ Correct Answer Explanation

Integrating f(x) = 2x gives the family of curves y = x² + C, where C represents a vertical shift. Comparing the given curve y = x² − 1 with the general form immediately shows that C = −1. Hence, Option C is correct. Options A, B, and D represent different members of the same family but do not match the given equation.

❓ Why Other Options Are Incorrect

Option A) C = 0 → This gives the curve y = x², whose vertex is at (0, 0). It does not represent the given graph y = x² − 1.
Option B) C = 1 → This produces y = x² + 1, which is shifted upward by one unit instead of downward by one unit.
Option D) C = 2 → This gives y = x² + 2, shifting the parabola upward by two units. It does not correspond to the given equation.

🎯 Strategy

Used: Option Grouping

Application: Compare the given equation directly with the general form y = x² + C. The constant term alone determines the correct member of the family.

Final Logic: Since x² + C = x² − 1, the required constant is −1.

🧠 Memory Trick

"Constant = Vertical Shift."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Family of Curves and Arbitrary Constant

6 The family of curves derived from integrating f(x) = 2x are parabolas. Based on the geometric interpretation of adding constant C, where does the vertex of these parabolas translate?

A Laterally on the x-axis

B Vertically on the y-axis

C Parallel to the x-axis

D Through an unconstrained, random parabolic path

💡 Short Explanation

The constant C changes only the function value. The x-coordinate of the vertex remains unchanged. Hence, the parabola shifts vertically along the y-axis.

✅ Correct Answer Explanation

The antiderivative of 2x is y = x² + C. The constant C adds or subtracts a fixed value from every point on the graph, producing a vertical translation without changing the shape or the x-coordinate of the vertex. Therefore, the vertex moves only along the y-axis. Thus, Option B is correct. The remaining options incorrectly describe the effect of the integration constant.

❓ Why Other Options Are Incorrect

Option A) Laterally on the x-axis → A lateral (horizontal) shift would require replacing x with (x − h). Changing C does not alter the x-coordinate of any point.
Option C) Parallel to the x-axis → This statement is misleading. The graph shifts upward or downward, which is a movement parallel to the y-axis, not the x-axis.
Option D) Through an unconstrained, random parabolic path → The constant C creates a systematic vertical translation. The movement is neither random nor changes the parabola's shape.

🎯 Strategy

Used: Elimination

Application: Recall that adding a constant affects only the output (y-value). Eliminate options involving horizontal or random movement.

Final Logic: Since C changes only the y-values, the parabola translates vertically, making Option B correct.

🧠 Memory Trick

"Add C → Move Up or Down."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Geometrical Meaning of the Constant of Integration (Family of Curves)

7 Evaluate the standard algebraic integral by separating terms:
\(\int \frac{x^{2}+1}{x^{2}} dx.\)

A \(x-\frac{1}{x}+C\)

B \(x+\frac{1}{x}+C\)

C \(\frac{x^{3}}{3}+x+C\)

D \(1-\frac{1}{x^{2}}+C\)

💡 Short Explanation

Rewrite the integrand as 1 + 1/x². Integrate each term separately. Apply the power rule and include the constant C.

✅ Correct Answer Explanation

First simplify the integrand: \(\frac{x^{2}+1}{x^{2}}=1+\frac{1}{x^{2}}=1+x^{-2}.\) Now integrate term by term: \(\int 1 dx=x,\int x^{-2}dx=-x^{-1}=-\frac{1}{x}.\) Hence, \(\int \frac{x^{2}+1}{x^{2}}dx=x-\frac{1}{x}+C.\) Therefore, Option A is correct. Options B, C, and D arise from incorrect application of standard integration rules.

❓ Why Other Options Are Incorrect

Option B) x + 1/x + C → The integral of x⁻² is −x⁻¹, not +x⁻¹. The negative sign has been omitted.
Option C) x³ / 3 + x + C → This incorrectly treats the integrand as x² + 1 instead of first simplifying the fraction by dividing by x².
Option D) 1 - 1/x² + C → This is not the antiderivative. It neither integrates the constant term correctly nor applies the power rule to x⁻².

🎯 Strategy

Used: Elimination

Application: Simplify the rational expression before integrating. Eliminate options that ignore simplification or incorrectly apply the power rule.

Final Logic: After rewriting the integrand as 1 + x⁻², direct integration gives x − 1/x + C, making Option A correct.

🧠 Memory Trick

"Simplify first, integrate next."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Standard Indefinite Integrals and Algebraic Simplification

8 Using substitution rules where necessary, evaluate the following related integral:
\(\int e^{2x+3} dx.\)

A \(2e^{2x+3}+C\)

B \(\frac{1}{2}e^{2x+3}+C\)

C \(e^{2x+3}+C\)

D \(\frac{1}{3}e^{2x+3}+C\)

💡 Short Explanation

Let u = 2x + 3. Then du = 2dx, so dx = du/2. Integrate eᵘ and substitute back.

✅ Correct Answer Explanation

Use substitution by taking u = 2x + 3. Then du = 2dx, giving dx = du/2. Therefore, \(\int e^{2x+3}dx=\frac{1}{2}\int e^{u} du=\frac{1}{2}e^{u}+C=\frac{1}{2}e^{2x+3}+C.\) Hence, Option B is correct. Options A, C, and D use incorrect multiplying factors and ignore the derivative of the exponent.

❓ Why Other Options Are Incorrect

Option A) 2 e^(2x+3) + C → The factor should be 1/2, not 2. The derivative of the exponent must be divided out during integration.
Option C) e^(2x+3) + C → This ignores the chain rule. Since the exponent has derivative 2, a factor of 1/2 is required.
Option D) ⅓ e^(2x+3) + C → The denominator should equal the derivative of the exponent, which is 2, not 3.

🎯 Strategy

Used: Substitution

Application: Replace the linear expression in the exponent with a new variable and adjust dx using its derivative before integrating.

Final Logic: Since the exponent's derivative is 2, divide by 2, giving ½e^(2x+3) + C.

🧠 Memory Trick

"Exponent derivative goes to the denominator."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Integration by Substitution (Exponential Functions)

9 Which of the following is the correctly evaluated expression for the standard integral
\(\int \frac{1}{a^{2}-x^{2}} dx?\)

A \(\frac{1}{2a}log⁡∣\frac{a+x}{a-x}∣+C\)

B \(\frac{1}{2a}log⁡∣\frac{x-a}{x+a}∣+C\)

C \({sin⁡}^{-1}\left(\frac{x}{a}\right)+C\)

D \(log⁡∣x+\sqrt{x^{2}-a^{2}}∣+C\)

💡 Short Explanation

Use the standard integral formula for 1/(a² − x²). The result is a logarithmic function. Include the constant of integration C.

✅ Correct Answer Explanation

The standard integral formula is \(\int \frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}log⁡∣\frac{a+x}{a-x}∣+C.\) This follows from partial fractions. Therefore, Option A is correct. Option B is the equivalent expression for ∫1/(x²−a²),dx, Option C corresponds to ∫1/\sqrt{a²-x²},dx, and Option D belongs to a different standard logarithmic integral.

❓ Why Other Options Are Incorrect

Option B) 1/(2a) log|(x-a)/(x+a)| + C → This is the standard result for ∫1/(x²−a²),dx, not for ∫1/(a²−x²),dx. The sign is reversed because the denominator is different.
Option C) sin⁻¹(x/a) + C → This is the antiderivative of 1/\sqrt{a²−x²}, not 1/(a²−x²). It represents a completely different standard formula.
Option D) log|x + √(x² - a²)| + C → This expression arises while integrating 1/\sqrt{x²−a²}, not the given rational function.

🎯 Strategy

Used: Option Grouping

Application: Identify the denominator first and match it with the corresponding standard integral formula instead of relying on similar-looking logarithmic expressions.

Final Logic: The denominator is a² − x², so the correct standard formula is given in Option A.

🧠 Memory Trick

"a² − x² → Log with (a + x)/(a − x)."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Standard Integral Formulae (Special Forms)

10 Evaluate the standard integral formula:
\(\int \sqrt{x^{2}+a^{2}} dx.\)

A \(\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}log⁡∣x+\sqrt{x^{2}+a^{2}}∣+C\)

B \(\frac{x}{2}\sqrt{x^{2}-a^{2}}+\frac{a^{2}}{2}log⁡∣x+\sqrt{x^{2}-a^{2}}∣+C\)

C \(log⁡∣x+\sqrt{x^{2}+a^{2}}∣+C\)

D \(\frac{1}{a}{tan⁡}^{-1}\left(\frac{x}{a}\right)+C\)

💡 Short Explanation

Recognize the integral as a standard formula. The result contains both an algebraic and a logarithmic term. Add the constant of integration C.

✅ Correct Answer Explanation

The standard integral is \(\int \sqrt{x^{2}+a^{2}} dx=\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}log⁡∣x+\sqrt{x^{2}+a^{2}}∣+C.\) Hence, Option A is correct. Option B incorrectly replaces x²+a² with x²−a², Option C is incomplete because it omits the algebraic term, and Option D is the standard integral of 1/(a²+x²), not √(x²+a²).

❓ Why Other Options Are Incorrect

Option B) x/2 √(x² - a²) + a²/2 log|x + √(x² - a²)| + C → This corresponds to a different standard integral involving √(x²−a²) and cannot be used for the given integrand.
Option C) log|x + √(x² + a²)| + C → This includes only the logarithmic component and omits the essential algebraic term present in the correct antiderivative.
Option D) 1/a tan⁻¹(x/a) + C → This is the standard result for ∫dx/(a²+x²) and has no relation to integrating √(x²+a²).

🎯 Strategy

Used: Option Grouping

Application: Identify the exact form of the integrand and match it with the corresponding standard integral formula without confusing it with similar expressions.

Final Logic: Only Option A exactly matches the standard NCERT formula for ∫√(x²+a²),dx.

🧠 Memory Trick

"Root (x² + a²) → Half algebraic term + Half log term."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Standard Integral Formulae (Special Integrals)

11 By applying the fundamental property that integration and differentiation are inverse operators, evaluate
\(\int \frac{d}{dx}(f(x)) dx.\)

A \(f^{'}(x)\)

B \(f(x)+C\)

C \(f(x)\)

D \(\frac{d}{dx}[f(x)]+C\)

💡 Short Explanation

Differentiation and integration are inverse operations. Integrating a derivative restores the original function. Include the arbitrary constant C in indefinite integration.

✅ Correct Answer Explanation

Since differentiation and integration are inverse processes, \(\int \frac{d}{dx}[f(x)] dx=\int f^{'}(x) dx=f(x)+C.\) The constant C is necessary because the derivative of every constant is zero. Therefore, all functions differing only by a constant have the same derivative. Hence, Option B is correct. Options A, C, and D either omit the integration result or fail to include the arbitrary constant.

❓ Why Other Options Are Incorrect

Option A) f'(x) → This is the derivative of the function, not the result obtained after integrating the derivative. Integration reverses differentiation and returns the original function plus a constant.
Option C) f(x) → Although the original function is obtained, the arbitrary constant C is missing. Every indefinite integral must include the constant of integration.
Option D) d/dx [f(x)] + C → This simplifies to f'(x) + C, which is not the antiderivative of f'(x). It incorrectly leaves the differentiation operator unchanged.

🎯 Strategy

Used: Elimination

Application: Recall that integration and differentiation cancel each other except for the constant of integration. Eliminate options that omit C or retain the derivative.

Final Logic: The inverse property gives f(x) + C, making Option B the correct answer.

🧠 Memory Trick

"Integrate a derivative → Original + C."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Integration as the Inverse Process of Differentiation

12 Applying the linear property of scalar multiplication in integrals, if I = ∫ k f(x) dx (where k is a constant), which equivalent expression is correct?

A k ∫ f(x) dx

B ∫ f(kx) dx

C k + ∫ f(x) dx

D k x ∫ f(x) dx

💡 Short Explanation

A constant factor can be taken outside the integral. This is called the linearity property of integration. The property holds only when k is constant.

✅ Correct Answer Explanation

The linearity property of integration states that a constant multiplier may be taken outside the integral: \(\int kf(x) dx=k\int f(x) dx,\) where k is a constant. This simplifies calculations and is one of the fundamental properties of indefinite integrals. Therefore, Option A is correct. Options B, C, and D incorrectly modify the function or misuse the constant factor.

❓ Why Other Options Are Incorrect

Option B) ∫ f(kx) dx → Multiplying the function by a constant is not the same as replacing x with kx. This changes the function and requires substitution.
Option C) k + ∫ f(x) dx → A constant multiplier cannot be converted into an added constant. This violates the linearity property of integration.
Option D) k x ∫ f(x) dx → Multiplying by x introduces a variable factor that is not part of the given property. The constant alone should be taken outside the integral.

🎯 Strategy

Used: Option Grouping

Application: Recall the standard properties of indefinite integrals and identify the expression that preserves the constant multiplier without changing the function.

Final Logic: A constant factor comes outside the integral unchanged, so Option A is correct.

🧠 Memory Trick

"Constant Out, Integral Stays."

📚 Class XII – NCERT Mathematics Part II – Chapter: Integrals – Topic: Properties of Indefinite Integrals (Linearity Property)

13 Evaluate the expanded polynomial expression:
\(\int (x+3)(x+2) dx.\)

A \(\frac{x^{3}}{3}+\frac{5x^{2}}{2}+6x+C\)

B \(x^{2}+5x+6+C\)

C \(\frac{x^{3}}{3}+\frac{5x}{2}+6+C\)

D \(\frac{{\left(x+3\right)}^{2}}{2}+C\)

💡 Short Explanation

Expand the polynomial first. Integrate each term separately using the power rule. Add the constant of integration after completing the integration.

✅ Correct Answer Explanation

Expand the integrand: \((x+3)(x+2)=x^{2}+5x+6\) Now integrate term by term: \(\int (x^{2}+5x+6) dx=\frac{x^{3}}{3}+\frac{5x^{2}}{2}+6x+C\) Hence Option A is correct. Option B gives the expanded polynomial instead of its integral. Option C integrates the middle and constant terms incorrectly. Option D incorrectly treats the expression as a single square instead of expanding and integrating.

❓ Why Other Options Are Incorrect

Option B → x² + 5x + 6 + C

This is merely the expanded integrand, not the integral. Integration increases the power of each variable term by one and divides by the new exponent.

Option C → x³ / 3 + 5x / 2 + 6 + C

The first term is correct, but the remaining terms are integrated incorrectly. The integral of \(5x\) is \(\frac{5x^{2}}{2}\), and the integral of 6 is \(6x\).

Option D → (x + 3)² / 2 + C

This incorrectly assumes the integrand is directly related to \({\left(x+3\right)}^{2}\). The product must first be expanded before applying integration.

🎯 Strategy

Used: Substitution (Expand First Method)

Application: First simplify the algebraic expression by expansion. Then integrate each term individually using the standard power rule.

Final Logic: Expand → Integrate each term → Add C.

🧠 Memory Trick

"Expand → Integrate → Constant (EIC)."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration of Polynomial Functions using Power Rule

14 How can the improper rational function
\(\frac{x^{2}+7x+12}{x^{2}+3x+2}\)
be rewritten for integration using algebraic long division?

A \(1+\frac{4x+10}{x^{2}+3x+2}\)

B \(x+\frac{4x+10}{x^{2}+3x+2}\)

C \(1+(x^{2}+7x+12)\)

D \(2+\frac{4x}{x^{2}+3x+2}\)

💡 Short Explanation

Degrees of numerator and denominator are equal. Apply algebraic long division first. Express the function as quotient plus remainder over denominator before integration.

✅ Correct Answer Explanation

Since the numerator and denominator have the same degree, perform algebraic long division: \(x^{2}+7x+12=(x^{2}+3x+2)+(4x+10)\) Therefore, \(\frac{x^{2}+7x+12}{x^{2}+3x+2}=1+\frac{4x+10}{x^{2}+3x+2}\) Thus Option A is correct. Option B gives an incorrect quotient. Option C is not a valid decomposition. Option D has both an incorrect quotient and remainder.

❓ Why Other Options Are Incorrect

Option B → x + (4x + 10) / (x² + 3x + 2)

The quotient from long division is 1, not x. Multiplying the denominator by \(x\) would produce a cubic expression, which is inconsistent.

Option C → 1 + (x² + 7x + 12)

This ignores the denominator completely. Long division always leaves the remainder divided by the original denominator.

Option D → 2 + (4x) / (x² + 3x + 2)

Both the quotient and remainder are incorrect. The correct remainder after subtraction is \(4x+10\), not \(4x\).

🎯 Strategy

Used: Elimination

Application: Check whether each option satisfies the identity:

Numerator = (Quotient × Denominator) + Remainder.
Only one option reconstructs the original numerator exactly.

Final Logic: Correct long division gives Quotient = 1 and Remainder = 4x + 10.

🧠 Memory Trick

"Same degree? Divide first, integrate later."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration of Rational Functions using Algebraic Division

15 The marginal revenue function for a specific output is
\(MR=40-10x^{2}.\)
If the firm's total revenue is ₹120 at an output level of 3 units, determine the exact total revenue function \(R(x)\).

A \(R(x)=40x-\frac{10x^{3}}{3}+90\)

B \(R(x)=40x-10x^{3}+120\)

C \(R(x)=40x-\frac{10x^{3}}{3}\)

D \(R(x)=40-20x+90\)

💡 Short Explanation

Integrate the marginal revenue function. Use the given revenue value to determine the integration constant. Substitute the constant into the general solution to obtain the particular revenue function.

✅ Correct Answer Explanation

Marginal revenue is the derivative of total revenue: \(MR=\frac{dR}{dx}=40-10x^{2}\) Integrating, \(R(x)=40x-\frac{10x^{3}}{3}+C\) Using the condition \(R(3)=120\): \(120=40(3)-\frac{10(27)}{3}+C120=120-90+CC=90\) Hence, \(R(x)=40x-\frac{10x^{3}}{3}+90\) Therefore, Option A is correct. Option B integrates incorrectly, Option C omits the constant, and Option D is not obtained by integration.

❓ Why Other Options Are Incorrect

Option B → R(x) = 40x - 10x³ + 120

The integral of \(-10x^{2}\)is \(-\frac{10x^{3}}{3}\), not \(-10x^{3}\). The integration constant is also incorrect.

Option C → R(x) = 40x - 10x³ / 3

This is only the general integral without applying the condition \(R(3)=120\). The particular solution must include the calculated constant.

Option D → R(x) = 40 - 20x + 90

This expression is unrelated to integrating the marginal revenue function and does not differentiate back to \(40-10x^{2}\).

🎯 Strategy

Used: Substitution

Application: Integrate the marginal function first, then substitute the given revenue value to determine the unknown constant.

Final Logic: Integrate → Use \(R(3)=120\)→ Obtain \(C=90\).

🧠 Memory Trick

"Integrate first, substitute next, constant last."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Particular Integrals and Applications of Integration

16 The marginal cost of producing \(x\) units is given by
\(MC=17+\frac{200}{x+1}.\)
Find the explicit total cost function \(C(x)\)if the daily fixed cost is ₹2000.

A \(C(x)=17x+200log⁡∣x+1∣+2000\)

B \(C(x)=17x^{2}+200log⁡∣x+1∣+2000\)

C \(C(x)=17x+\frac{200}{x}+2000\)

D \(C(x)=17+200log⁡∣x+1∣+2000\)

💡 Short Explanation

Integrate each marginal cost term separately. The logarithmic rule applies to \(1/(x+1)\). Fixed cost becomes the integration constant in the total cost function.

✅ Correct Answer Explanation

Marginal cost is the derivative of total cost: \(MC=\frac{dC}{dx}=17+\frac{200}{x+1}\) Integrating, \(C(x)=17x+200log⁡∣x+1∣+K\) Since the fixed cost is ₹2000, \(K=2000\) Thus, \(C(x)=17x+200log⁡∣x+1∣+2000\) Hence Option A is correct. Option B integrates 17 incorrectly. Option C incorrectly integrates the rational term. Option D omits the integration of the constant 17.

❓ Why Other Options Are Incorrect

Option B → C(x) = 17x² + 200 log|x + 1| + 2000

The integral of 17 is \(17x\), not \(17x^{2}\). The power rule has been applied incorrectly.

Option C → C(x) = 17x + 200/x + 2000

The integral of \(\frac{1}{x+1}\)is \(log⁡∣x+1∣\), not \(\frac{1}{x}\). This violates the standard logarithmic integration rule.

Option D → C(x) = 17 + 200 log|x + 1| + 2000

The constant 17 should integrate to \(17x\). Leaving it unchanged results in an incorrect total cost function.

🎯 Strategy

Used: Substitution

Application: Recognize the standard logarithmic integral \(\int \frac{1}{x+a} dx=log⁡∣x+a∣+C\), then add the fixed cost as the integration constant.

Final Logic: Integrate both terms correctly and replace the constant with the given fixed cost of ₹2000.

🧠 Memory Trick

"One over linear gives logarithm."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration of Rational Functions and Applications to Cost Functions

📄 Passage

In applied mathematics, finding the particular integral from an indefinite integral requires evaluating the arbitrary constant 'C' using specific real-world data points. For instance, the total revenue function R(x) is found by integrating the marginal revenue (MR) function with respect to the number of units sold, x. The resulting function initially contains the unknown constant 'C'. If a company knows its total revenue at a specific sales volume, such as R(2) = 240, this condition acts as the key to unlocking the exact revenue model. By substituting x = 2 and equating the entire expression to 240, the firm can solve algebraically for 'C'. Substituting this verified constant back into the equation provides the particular integral, enabling accurate predictions.

If evaluating the integral of marginal revenue gives R(x) = 80x + 10x² + x³ + C, and substituting x = 2 gives a known revenue of 240, what algebraic equation must be solved to find C?

A 80(2) + 10(2)² + (2)³ + C = 240

B 80 + 10(2) + 3 + C = 240

C 80x + 10x² + x³ = 240

D C = 240 × 2

💡 Short Explanation

Substitute the given value of \(x\) into the integrated function. Replace the function value with the known revenue. Solve the resulting equation to determine the constant \(C\).

✅ Correct Answer Explanation

The particular integral is obtained by using the given condition \(R(2)=240\). Substitute \(x=2\) into \(R(x)=80x+10x^{2}+x^{3}+C\) to obtain \(80(2)+10(2)^{2}+(2)^{3}+C=240.\) This equation is solved for \(C\). Therefore, Option A is correct. Option B substitutes incorrectly, Option C omits the constant and substitution, while Option D has no mathematical basis.

❓ Why Other Options Are Incorrect

Option B → 80 + 10(2) + 3 + C = 240

The substitution is performed incorrectly. The terms \(80x\), \(10x^{2}\), and \(x^{3}\)must each be evaluated at \(x=2\), giving 160, 40, and 8 respectively.

Option C → 80x + 10x² + x³ = 240

This leaves the variable \(x\) unchanged and ignores the integration constant \(C\). The given condition requires substituting \(x=2\).

Option D → C = 240 × 2

The constant is not obtained by multiplying the revenue by two. It is found only after substituting into the complete revenue equation.

🎯 Strategy

Used: Substitution

Application: Insert the given value of the independent variable into the integrated function and equate it to the known function value.

Final Logic: Known value + substitution = equation for \(C\).

🧠 Memory Trick

"Substitute first, solve C next."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Particular Integral Using Initial Condition

📄 Passage

What characteristic explicitly defines a "particular integral" as opposed to an indefinite integral in the context of business applications?

A It contains no variables, only fixed values.

B The constant 'C' has been specifically solved for and replaced with a numerical value.

C It represents an infinite family of parallel market curves.

D It relies solely on differentiation rather than integration.

💡 Short Explanation

An indefinite integral contains an arbitrary constant. A particular integral has a fixed numerical constant. The constant is determined using the given condition or initial value.

✅ Correct Answer Explanation

An indefinite integral represents a family of functions because it contains the arbitrary constant \(C\). When additional information such as \(R(2)=240\) is available, the constant is evaluated and replaced with a specific numerical value. The resulting expression is called the particular integral. Hence, Option B is correct. Options A, C, and D do not describe the defining feature of a particular integral.

❓ Why Other Options Are Incorrect

Option A → It contains no variables, only fixed values.

A particular integral still contains variables such as \(x\). Only the arbitrary constant is replaced with a numerical value.

Option C → It represents an infinite family of parallel market curves.

An infinite family of curves corresponds to an indefinite integral, not a particular integral.

Option D → It relies solely on differentiation rather than integration.

A particular integral is obtained by integration followed by using a given condition. Differentiation is not the defining process.

🎯 Strategy

Used: Elimination

Application: Recall the defining property of a particular integral and eliminate statements that describe indefinite integrals or unrelated concepts.

Final Logic: A particular integral is uniquely identified by its numerical value of \(C\).

🧠 Memory Trick

"Particular = Particular C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Indefinite and Particular Integrals

19 Evaluate the integral using the substitution method:
\(\int \frac{2x}{1+x^{2}} dx\)
(Hint: Let \(t=1+x^{2}\).)

A \(log⁡∣1+x^{2}∣+C\)

B \(\frac{1+x^{2}}{2}+C\)

C \(2log⁡∣1+x^{2}∣+C\)

D \(x^{2}+C\)

💡 Short Explanation

Let \(t=1+x^{2}\). Then \(dt=2x dx\). The integral becomes \(\int \frac{1}{t}dt=log⁡∣t∣+C\), giving the required answer.

✅ Correct Answer Explanation

Using substitution, \(t=1+x^{2},dt=2x dx.\) Therefore, \(\int \frac{2x}{1+x^{2}} dx=\int \frac{1}{t} dt=log⁡∣t∣+C.\) Substituting back, \(log⁡∣1+x^{2}∣+C.\) Hence Option A is correct. Option B does not result from logarithmic integration. Option C incorrectly doubles the logarithm. Option D ignores the denominator completely.

❓ Why Other Options Are Incorrect

Option B → (1 + x²) / 2 + C

This is not obtained from integrating \(\frac{1}{1+x^{2}}\)with its derivative in the numerator. The standard logarithmic substitution rule has not been applied.

Option C → 2 log|1 + x²| + C

The factor 2 disappears because \(dt=2x dx\). No extra coefficient remains after substitution.

Option D → x² + C

Differentiating \(x^{2}+C\) gives \(2x\), which does not match the original integrand containing the denominator \(1+x^{2}\).

🎯 Strategy

Used: Substitution

Application: Recognize that the numerator is the derivative of the denominator, making substitution the quickest method.

Final Logic: Derivative of denominator present → Use substitution → Obtain logarithm.

🧠 Memory Trick

"Derivative below, log will grow."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration by Substitution

20 In an equilibrium application problem, what mathematical step immediately follows equating the quantity demanded (xd) to the quantity supplied (xs)?

A Substituting the arbitrary constant C

B Solving the equation for the equilibrium price p

C Integrating both demand and supply functions

D Subtracting the market fixed cost

💡 Short Explanation

Market equilibrium occurs when demand equals supply. Set \(x_{d}=x_{s}\). Solve the resulting equation to determine the equilibrium price before finding the equilibrium quantity.

✅ Correct Answer Explanation

In equilibrium analysis, \(x_{d}=x_{s}.\) After equating the demand and supply functions, the next step is solving the equation for the unknown market price \(p\). Once the equilibrium price is found, it can be substituted into either equation to obtain the equilibrium quantity. Therefore, Option B is correct. Options A, C, and D are unrelated to the standard equilibrium procedure.

❓ Why Other Options Are Incorrect

Option A → Substituting the arbitrary constant C

The constant of integration appears only in integration problems and has no role in solving market equilibrium equations.

Option C → Integrating both demand and supply functions

Equilibrium problems generally involve solving algebraic equations, not integrating demand and supply functions.

Option D → Subtracting the market fixed cost

Fixed cost is associated with production and cost analysis, not with determining market equilibrium.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recognize the standard sequence in equilibrium problems: equate demand and supply, solve for price, then determine quantity.

Final Logic: Demand = Supply → Solve for price → Find equilibrium quantity.

🧠 Memory Trick

"Equal first, Price next."

📚 Class XII – NCERT Mathematics – Chapter: Applications of Integrals – Topic: Economic Applications and Market Equilibrium Concepts

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