CUET UG Applied Mathematics Booster Test 3 - Integration by Parts and Definite Integrals -

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CUET UG Applied Mathematics Booster Test 3 - Integration by Parts and Definite Integrals

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 3 - Integration by Parts and Definite Integrals

1 / 20

QUESTION 1 OF 20

By mathematically reversing the continuous Product Rule of differentiation
\(\frac{d}{dx}[u(x)v(x)]=u(x)v^{'}(x)+v(x)u^{'}(x),\)
what is the explicitly isolated analytical integral form mapped to
\(\int u(x)v^{'}(x) dx\)
strictly before any variables are redefined as \(f(x)\)and \(g(x)\)?

Answer Review

1 By mathematically reversing the continuous Product Rule of differentiation
\(\frac{d}{dx}[u(x)v(x)]=u(x)v^{'}(x)+v(x)u^{'}(x),\)
what is the explicitly isolated analytical integral form mapped to
\(\int u(x)v^{'}(x) dx\)
strictly before any variables are redefined as \(f(x)\)and \(g(x)\)?

A \(\int u(x)v(x) dx-v(x)u^{'}(x)\)

B \(u^{'}(x)v^{'}(x)+\int u(x)v(x) dx\)

C \(u(x)v(x)-\int v(x)u^{'}(x) dx\)

D \(\frac{u(x)v^{'}(x)}{\int v(x)u^{'}(x) dx}\)

💡 Short Explanation

Start with the Product Rule. Integrate both sides. Rearrange the equation.

✅ Correct Answer Explanation

Integrating the Product Rule gives \(\int u(x)v^{'}(x) dx=u(x)v(x)-\int v(x)u^{'}(x) dx,\) which is the Integration by Parts formula.

❓ Why Other Options Are Incorrect

Option A: Not obtained from integrating the Product Rule.
Option B: Incorrect mathematical identity.
Option D: Not a valid integration formula.

🎯 Strategy

Used

Elimination

Application: Recall the derivation of Integration by Parts.

Final Logic: Integration of the Product Rule gives the required expression.

🧠 Memory Trick

Product Rule → Integrate → Rearrange

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Integration by Parts

2 Match the highly specific complex rational/exponential integrand variations listed in List I with their precisely calculated evaluated integral answers in List II.

List I	List II
1. — \(\int e^{x}\left(\frac{1}{x},\ \frac{1}{x^{2}}\right) dx\)	a. — \(\frac{x}{\log\,x}+C\)
2. — \(\int e^{x}\left(log⁡x+\frac{1}{x}\right) dx\)	b. — \(\frac{e^{x}}{x}+C\)
3. — \(\int \left(\frac{1}{\log\,x},\ \frac{1}{{\left(log⁡x\right)}^{2}}\right) dx\)	c. — \(e^{x}log⁡x+C\)
4. — \(\int e^{x}\left(\frac{x-1}{x^{2}}\right) dx\)	d. — \(\frac{e^{x}}{x}+C\)

A (1)-(d), (2)-(c), (3)-(a), (4)-(b)

B (1)-(a), (2)-(b), (3)-(c), (4)-(d)

C (1)-(b), (2)-(d), (3)-(a), (4)-(c)

D (1)-(c), (2)-(a), (3)-(d), (4)-(b)

💡 Short Explanation

Recall the standard Integration by Parts results. Match each integral correctly. Select the correct correspondence.

✅ Correct Answer Explanation

The correct matching is: 1 → d 2 → c 3 → a 4 → b Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B: Incorrect correspondence.
Option C: Several mappings are interchanged.
Option D: Does not match the standard results.

🎯 Strategy

Used

Option Grouping

Application: Match each integral with its evaluated result.

Final Logic: Only Option A gives the correct mapping.

🧠 Memory Trick

Recognize Standard Results

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Integration by Parts

3 When analytically evaluating single-term non-algebraic integrands like
\(\int log⁡x dxor\int {sin⁡}^{-1}(x) dx\)
using explicit Integration by Parts, which of the following technical procedural statements apply correctly?
A. The number \(1\) is mathematically introduced as a valid algebraic second function \(g(x)\).
B. The isolated logarithmic or inverse trigonometric term is rigidly taken as the first function \(f(x)\).
C. The integral of the second function accurately evaluates identically to \(x\).
D. The ILATE rule breaks down and must be universally abandoned for these cases.

A A, D

B A, B

C B, C, D

D A, B, C

💡 Short Explanation

Introduce \(1\) as the second function. Apply ILATE. Integrate \(1\).

✅ Correct Answer Explanation

Statements A, B and C are correct. Statement D is incorrect because ILATE remains applicable.

❓ Why Other Options Are Incorrect

Option A: Omits B and C.
Option B: Omits C.
Option C: Includes incorrect statement D.

🎯 Strategy

Used

Elimination

Application: Verify each statement separately.

Final Logic: Only A, B and C are correct.

🧠 Memory Trick

Log First, One Second

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Choosing Functions in Integration by Parts

4 Which definitive technical statement surrounding the theoretical edges of the ILATE function selection algorithm is mathematically INCORRECT?

A Selecting an inverse trigonometric function as the second function is usually avoided because its direct integral is not readily available.

B The absolute rigid order of ILATE ensures that the first function always mathematically zeroes out to a constant after two derivatives.

C The sequence is designed primarily to ensure the secondary integral \(\int f^{'}(x)\left[\int g(x) dx\right]dx\) evaluates simpler than the original equation.

D The rule relies on the practical observation that exponential functions are the easiest to integrate repeatedly without escalating complexity.

💡 Short Explanation

ILATE is a guideline. It is not an absolute rule. The statement is too rigid.

✅ Correct Answer Explanation

ILATE helps choose functions efficiently but does not guarantee that the first function becomes constant after two differentiations.

❓ Why Other Options Are Incorrect

Option A: Correct application of ILATE.
Option C: Correct objective of the method.
Option D: Correct practical observation.

🎯 Strategy

Used

Extreme Word Filter

Application: Look for absolute words like "always."

Final Logic: Absolute claims are generally incorrect.

🧠 Memory Trick

ILATE = Guide, Not Rule

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: ILATE Rule

5 Execute the highly complex rational breakdown of exponential parameters by evaluating
\(\int e^{x}\left(\frac{x-1}{x^{2}}\right) dx.\)

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

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

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

D \(e^{x}log⁡x+C\)

💡 Short Explanation

Rewrite the fraction. Apply the standard identity. Simplify.

✅ Correct Answer Explanation

Since \(\frac{x-1}{x^{2}}=\frac{1}{x}-\frac{1}{x^{2}},\) using the standard result, \(\int e^{x}\left(\frac{1}{x},\ \frac{1}{x^{2}}\right)dx=\frac{e^{x}}{x}+C.\)

❓ Why Other Options Are Incorrect

Option A: Does not differentiate to the integrand.
Option B: Represents only part of the required expression.
Option D: Incorrect antiderivative.

🎯 Strategy

Used

Substitution

Application: Rewrite the integrand into a standard form.

Final Logic: Match the transformed integrand with the standard result.

🧠 Memory Trick

Split First → Standard Result

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Standard Integration by Parts Results

6 Calculate the continuous non-linear fraction evaluation heavily utilized for polynomial reduction:
\(\int e^{x}\left(\frac{x^{2}+1}{{\left(x+1\right)}^{2}}\right) dx.\)
(Hint: Constrain to standard form by adding and subtracting terms.)

A \(e^{x}\left(\frac{1}{{\left(x+1\right)}^{2}}\right)+C\)

B \(e^{x}\left(\frac{x+1}{x-1}\right)+C\)

C \(e^{x}\left(\frac{x-1}{x+1}\right)+C\)

D \(e^{x}\left(\frac{x}{x+1}\right)+C\)

💡 Short Explanation

Rewrite the rational expression. Apply the standard Integration by Parts identity. Simplify.

✅ Correct Answer Explanation

By expressing \(\frac{x^{2}+1}{{\left(x+1\right)}^{2}}\) in a suitable standard form and applying the known result, \(\int e^{x}\left(\frac{x^{2}+1}{{\left(x+1\right)}^{2}}\right) dx=e^{x}\left(\frac{x-1}{x+1}\right)+C.\)

❓ Why Other Options Are Incorrect

Option A: Incomplete result.
Option B: Incorrect rational expression.
Option D: Does not satisfy differentiation.

🎯 Strategy

Used

Substitution

Application: Rewrite the integrand into a standard form before integration.

Final Logic: Simplification followed by the standard result gives the answer.

🧠 Memory Trick

Rewrite First → Integrate Later

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Integration by Parts

7 Structurally derive the definitive equivalent mathematical expansion utilized explicitly to integrate the fractional logarithmic compound
\(\int \left(\frac{1}{\log\,x},\ \frac{1}{{\left(log⁡x\right)}^{2}}\right) dx.\)
(Hint: Substitute \(t=log⁡x\).)

A \(\frac{x}{\log\,x}+C\)

B \(xlog⁡x+C\)

C \(-\frac{1}{{\left(log⁡x\right)}^{2}}+C\)

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

💡 Short Explanation

Substitute \(t=log⁡x\). Simplify the integral. Apply the standard result.

✅ Correct Answer Explanation

Using the substitution \(t=log⁡x,\) the integral simplifies to the standard form, giving \(\frac{x}{\log\,x}+C.\)

❓ Why Other Options Are Incorrect

Option B: Incorrect antiderivative.
Option C: Only part of the expression is considered.
Option D: Incorrect exponential form.

🎯 Strategy

Used

Substitution

Application: Transform the logarithmic expression into a standard integral.

Final Logic: The substitution leads directly to the required result.

🧠 Memory Trick

Log → Substitute

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Logarithmic Integrals

8 Analyze and evaluate the continuously decaying logarithmic integral natively expressed as
\(\int x^{2}(log⁡x)^{2} dx.\)

A \(\frac{{\left(log⁡x\right)}^{3}}{3x}+C\)

B \(\frac{x^{3}}{3}(log⁡x)^{2}-\frac{2x^{3}}{9}log⁡x+\frac{2x^{3}}{27}+C\)

C \(x^{3}(log⁡x)^{2}-x^{3}log⁡x+C\)

D \(\frac{x^{3}}{3}(log⁡x)^{2}-\frac{2x^{3}}{9}+C\)

💡 Short Explanation

Apply Integration by Parts twice. Reduce the logarithmic power. Simplify.

✅ Correct Answer Explanation

Repeated Integration by Parts gives \(\frac{x^{3}}{3}(log⁡x)^{2}-\frac{2x^{3}}{9}log⁡x+\frac{2x^{3}}{27}+C.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect formula.
Option C: Missing required coefficients.
Option D: Incomplete expression.

🎯 Strategy

Used

Substitution

Application: Reduce the logarithmic power step by step.

Final Logic: Apply Integration by Parts repeatedly.

🧠 Memory Trick

Power of Log ↓ Each Step

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Higher Power Logarithmic Integrals

9 Theoretically applying the ILATE ranking probability, if attempting to evaluate
\(\int xsin⁡x dx,\)
what ensures the algebraic function \(x\) diminishes in complexity rather than expanding indefinitely?

A Assigning \(x\) as the first function \(f(x)\)so its derivative evaluates predictably to \(1\).

B Assigning \(x\) as the second function \(g(x)\)so it integrates to \(\frac{x^{2}}{2}\).

C Multiplying both functions by \(e^{x}\).

D Substituting the trigonometric function with a logarithmic function.

💡 Short Explanation

Follow the ILATE rule. Differentiate the algebraic function. Reduce complexity.

✅ Correct Answer Explanation

Choosing \(x\) as the first function gives \(\frac{d}{dx}(x)=1,\) making the remaining integral simpler.

❓ Why Other Options Are Incorrect

Option B: Makes the integral more complicated.
Option C: No mathematical basis.
Option D: Invalid substitution.

🎯 Strategy

Used

Elimination

Application: Select the choice that reduces the algebraic degree.

Final Logic: Differentiating \(x\) simplifies the integral.

🧠 Memory Trick

Differentiate Algebraic, Integrate Trigonometric

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Integration by Parts

10 Systematically evaluate the intensely nested, dual-form composite integration mathematically structured as: \(\int \left[log⁡(log⁡x)+\frac{1}{{\left(log⁡x\right)}^{2}}\right]dx.\)

A \(e^{x}log⁡(log⁡x)+C\)

B \(log⁡(log⁡x)\div log⁡x+C\)

C \(xlog⁡(log⁡x)-\frac{x}{\log\,x}+C\)

D \(xlog⁡(log⁡x)+\frac{x}{\log\,x}+C\)

💡 Short Explanation

Apply Integration by Parts. Use logarithmic identities. Simplify.

✅ Correct Answer Explanation

Using Integration by Parts, \(\int \left[log⁡(log⁡x)+\frac{1}{{\left(log⁡x\right)}^{2}}\right]dx=xlog⁡(log⁡x)-\frac{x}{\log\,x}+C.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect exponential form.
Option B: Not the required antiderivative.
Option D: Incorrect sign.

🎯 Strategy

Used

Substitution

Application: Apply Integration by Parts with logarithmic simplification.

Final Logic: The resulting expression differentiates back to the given integrand.

🧠 Memory Trick

Nested Log → Parts

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Advanced Integration by Parts

11 By executing the exact definite property of standard limit mapping where
\(\int_{0}^{a}\,f(x) dx=\int_{0}^{a}\,f(a-x) dx,\)
determine the value of
\(\int_{0}^{1}\,\frac{log⁡(1-x)}{log⁡(1-x)+log⁡x} dx.\)

A \(\frac{1}{2}\)

B \(1\)

C \(2\)

D \(0\)

💡 Short Explanation

Apply the property \(f(x)=f(a-x)\). Add the two equivalent integrals. Solve for the required value.

✅ Correct Answer Explanation

Using the substitution \(x\rightarrow 1-x\), \(I=\int_{0}^{1}\,\frac{\log\,x}{log⁡x+log⁡(1-x)} dx.\) Adding both expressions, \(2I=\int_{0}^{1}\,1 dx=1.\) Hence, \(I=\frac{1}{2}.\)

❓ Why Other Options Are Incorrect

Option B: Gives the total integral instead of half.
Option C: Exceeds the interval value.
Option D: The integral is not zero.

🎯 Strategy

Used

Substitution

Application: Replace \(x\) with \(1-x\) and combine the integrals.

Final Logic: Symmetry gives \(2I=1\).

🧠 Memory Trick

Symmetry ⇒ Half

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Properties of Definite Integrals

12 Determine the evaluated outcome for the mathematically dense interval boundary mapping defined fundamentally as:
\(\int_{1}^{3}\,\frac{\sqrt[3]{x}}{\sqrt[3]{x}+\sqrt[3]{4-x}} dx.\)

A \(0\)

B \(2\)

C \(1\)

D \(3\)

💡 Short Explanation

Use the property \(f(x)=f(a+b-x)\). Add complementary integrals. Evaluate.

✅ Correct Answer Explanation

Let \(I=\int_{1}^{3}\,\frac{\sqrt[3]{x}}{\sqrt[3]{x}+\sqrt[3]{4-x}}dx.\) Replacing \(x\) by \(4-x\), \(2I=\int_{1}^{3}\,1 dx=2.\) Therefore, \(I=1.\)

❓ Why Other Options Are Incorrect

Option A: Integral is positive.
Option B: Equals the interval length.
Option D: Greater than the interval length.

🎯 Strategy

Used

Substitution

Application: Apply symmetry of definite integrals.

Final Logic: Complementary functions sum to 1.

🧠 Memory Trick

Complementary Fractions ⇒ Half Interval

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Properties of Definite Integrals

13 Conceptually bounding the rigorous analytical limitations of the First Fundamental Theorem, if an area is functionally modeled continuously below the x-axis entirely, how does the mathematical evaluated fixed numerical value manifest its orientation?

A The numerical area evaluates natively as a strictly negative mathematical quantity.

B The numerical area naturally remains strictly positive regardless of placement.

C The definite integral immediately fails to generate a fixed mathematical number.

D The upper and lower interval boundary limits algebraically multiply identically by zero.

💡 Short Explanation

Signed area is considered. Region below the \(x\)-axis is negative. Definite integrals measure signed area.

✅ Correct Answer Explanation

A definite integral represents signed area. Therefore, when the graph lies completely below the \(x\)-axis, the value of the definite integral is negative.

❓ Why Other Options Are Incorrect

Option B: Ignores signed area.
Option C: True only if positive and negative areas cancel.
Option D: The integral remains well-defined.

🎯 Strategy

Used

Elimination

Application: Recall the geometric meaning of definite integrals.

Final Logic: Area below the \(x\)-axis contributes negatively.

🧠 Memory Trick

Below Axis ⇒ Negative

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Geometrical Interpretation of Definite Integrals

14 Systematically evaluate the definitive interval limit splitting integral explicitly defined across a continuous absolute value modulus: \(\int_{0}^{4}\,∣x-2∣ dx.\)

A \(8\)

B \(4\)

C \(2\)

D \(0\)

💡 Short Explanation

Split the integral at \(x=2\). Evaluate each part separately. Add the results.

✅ Correct Answer Explanation

Since \(∣x-2∣=\left\{\begin{pmatrix}2-x, & 0\leq x\leq 2,\\ x-2, & 2\leq x\leq 4,\end{pmatrix}\right.\) the integral becomes \(\int_{0}^{2}\,(2-x) dx+\int_{2}^{4}\,(x-2) dx=2+2=4.\)

❓ Why Other Options Are Incorrect

Option A: Double the correct value.
Option C: Only one half of the area.
Option D: The positive areas do not cancel.

🎯 Strategy

Used

Elimination

Application: Split the modulus into two intervals.

Final Logic: Total area equals 4.

🧠 Memory Trick

Modulus ⇒ Split at Zero Point

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Integration of Modulus Functions

15 Calculate the numerically evaluated definite fractional integration result analytically computing out to: \(\int_{3}^{5}\,\frac{x^{2}}{\left(x-1)(x-2\right)} dx.\)

A \(2+\frac{1}{2}log⁡ \left(\frac{3}{2}\right)-4log⁡ \left(\frac{4}{3}\right)\)

B \(log⁡ \left(\frac{5}{3}\right)\)

C \(\frac{62}{5}-\frac{56}{3}\)

D \(2+log⁡ \left(\frac{3}{2}\right)+4log⁡ \left(\frac{4}{3}\right)\)

💡 Short Explanation

Apply partial fraction decomposition. Integrate each term. Substitute the limits.

✅ Correct Answer Explanation

After decomposing the rational function into partial fractions and evaluating between the limits 3 and 5, the result is \(2+\frac{1}{2}log⁡ ⁣\left(\frac{3}{2}\right)-4log⁡ ⁣\left(\frac{4}{3}\right).\)

❓ Why Other Options Are Incorrect

Option B: Incomplete evaluation.
Option C: Incorrect numerical expression.
Option D: Incorrect signs in the logarithmic terms.

🎯 Strategy

Used

Substitution

Application: Use partial fractions followed by definite integration.

Final Logic: Evaluate the antiderivative at the given limits.

🧠 Memory Trick

Partial Fractions → Logs → Limits

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Evaluation Using Partial Fractions

16 Calculate the continuous polynomial limits mapped mathematically evaluating the precise integration outcome: \(\int_{0}^{1}\,xe^{x} dx.\)

A \(e\)

B \(e-1\)

C \(1\)

D \(0\)

💡 Short Explanation

Apply Integration by Parts. Evaluate the antiderivative. Substitute the limits.

✅ Correct Answer Explanation

Using Integration by Parts, \(\int xe^{x} dx=xe^{x}-e^{x}+C.\) Applying the limits, \({\left[xe^{x},\ e^{x}\right]}_{0}^{1}=(e-e)-(-1)=1.\) Hence, the correct answer is Option C.

❓ Why Other Options Are Incorrect

Option A: Incorrect evaluation of the definite integral.
Option B: Omits part of the antiderivative.
Option D: The integral is not zero.

🎯 Strategy

Used

Substitution

Application: Evaluate the antiderivative at the given limits.

Final Logic: Apply \(F(1)-F(0)\).

🧠 Memory Trick

Parts → Apply Limits

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Evaluation Using Integration by Parts

📄 Passage

Equally important to the Consumers' Surplus is the related concept of Producers' Surplus (PS) established strictly in economic market models. The supply curve p = g(x) represents a highly reliable graphical representation mapping the exact relationship between the base price of a good and the continuous quantity heavily supplied by the firm. It mathematically shows the exact quantity of the commodity that producers will eagerly bring into the market at any given dynamic price p. If the prevailing market's exact equilibrium price evaluates numerically to p₀, there are often highly efficient producers operating in the background who would have been completely willing to securely supply the entire product yield at a baseline price much lower than p₀. The overall financial gain explicitly realized by these specific operating producers is collectively calculated and known identically as the Producers' Surplus. Thus,

Referencing the detailed formal text constraints defined directly above, what strict mathematical formula rigorously evaluates the bounded spatial geometric definition calculating exact Producers' Surplus (PS)?

A \(PS=\int_{0}^{x_{0}}\,g(x) dx-p_{0}x_{0}\)

B \(PS=p_{0}x_{0}+\int_{0}^{x_{0}}\,g(x) dx\)

C \(PS=\int_{0}^{x_{0}}\,p_{0}x_{0} dx\)

D \(PS=p_{0}x_{0}-\int_{0}^{x_{0}}\,g(x) dx\)

💡 Short Explanation

Area of rectangle. Subtract area under supply curve. Gives Producers' Surplus.

✅ Correct Answer Explanation

From the passage, \(PS=p_{0}x_{0}-\int_{0}^{x_{0}}\,g(x) dx.\) This is the standard formula for Producers' Surplus.

❓ Why Other Options Are Incorrect

Option A: Formula is reversed.
Option B: Areas are added instead of subtracted.
Option C: Incorrect mathematical model.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Match the formula directly from the passage.

Final Logic: The passage explicitly states the Producers' Surplus formula.

🧠 Memory Trick

Rectangle − Curve = PS

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Producers' Surplus

📄 Passage

To perfectly calculate and exact physical area derivations for Producers' Surplus defined above when dealing explicitly with the non-linear continuous supply function
\(p=x^{2}+4x+5,\)
set exactly at a static price of p₀ = 10, what specific integrated area calculation must be successfully subtracted from the rectangle?

A \(\int_{0}^{10}\,(x^{2}+4x+5) dx\)

B \(\int_{0}^{5}\,(x^{2}+4x+5) dx\)

C \(\int_{0}^{1}\,(x^{2}+4x+5) dx\)

D \(\int_{1}^{10}\,(x^{2}+4x+5) dx\)

💡 Short Explanation

Integrate the supply curve. Use the equilibrium quantity. Subtract from the rectangle.

✅ Correct Answer Explanation

At equilibrium, \(x^{2}+4x+5=10\) gives \(x_{0}=1\). Hence the area under the supply curve is \(\int_{0}^{1}\,(x^{2}+4x+5) dx.\)

❓ Why Other Options Are Incorrect

Option A: Uses price instead of quantity.
Option B: Incorrect upper limit.
Option D: Wrong interval.

🎯 Strategy

Used

Substitution

Application: Determine the equilibrium quantity before selecting limits.

Final Logic: Integrate from \(0\) to \(x_{0}\).

🧠 Memory Trick

Upper Limit = Quantity

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Producers' Surplus

19 Extrapolate and thoroughly evaluate the densely mapped geometric numerical definition explicitly bounding\(\int_{0}^{1}\,\frac{1}{\sqrt{1+x^{2}}} dx.\)

A \(\sqrt{2}-1\)

B \(\frac{\pi }{4}\)

C \(log⁡(1+\sqrt{2})\)

D \(log⁡(1-\sqrt{2})\)

💡 Short Explanation

Use the standard integral. Apply the limits. Simplify.

✅ Correct Answer Explanation

Using \(\int \frac{dx}{\sqrt{1+x^{2}}}=log⁡ ⁣\left(x,\ \sqrt{1+x^{2}}\right)+C,\) we get \({\left[log,\ ⁣\left(x,\ \sqrt{1+x^{2}}\right)\right]}_{0}^{1}=log⁡(1+\sqrt{2}).\)

❓ Why Other Options Are Incorrect

Option A: Incorrect evaluation.
Option B: Corresponds to another standard integral.
Option D: Invalid logarithmic value.

🎯 Strategy

Used

Substitution

Application: Recall the standard integral formula.

Final Logic: Evaluate the antiderivative at the limits.

🧠 Memory Trick

Root \(1+x^{2}\)→ Log Formula

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Standard Definite Integrals

20 Complete the analysis to directly evaluate the continuous numerical definite integral exclusively mapping boundaries dynamically:\(\int_{1}^{2}\,\frac{1}{x(1+log⁡x)^{2}} dx.\)

A \(log⁡(1+log⁡2)\)

B \(\frac{1}{1+log⁡2}\)

C \(e^{2}-e\)

D \(1-\frac{1}{1+log⁡2}\)

💡 Short Explanation

Use substitution. Change the limits. Evaluate.

✅ Correct Answer Explanation

Let \(t=1+log⁡x,dt=\frac{dx}{x}.\) Then, \(\int_{1}^{2}\,\frac{dt}{t^{2}}={\left[,\ \frac{1}{t}\right]}_{1}^{1+log⁡2}=1-\frac{1}{1+log⁡2}.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect antiderivative.
Option B: Only part of the final result.
Option C: Unrelated expression.

🎯 Strategy

Used

Substitution

Application: Replace the logarithmic expression with a new variable.

Final Logic: Evaluate using the transformed limits.

🧠 Memory Trick

Log Inside → Substitute

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Substitution Method in Definite Integrals

CUET UG Applied Mathematics Booster Test 3 - Integration by Parts and Definite Integrals

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