CUET UG Applied Mathematics Booster Test 2 - Integration by Parts and Definite Integrals -

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CUET UG Applied Mathematics Booster Test 2 - 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 2 - Integration by Parts and Definite Integrals

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QUESTION 1 OF 20

How does the fundamental differentiation equation
\(\frac{d}{dx}[u(x)v(x)]=u(x)v^{'}(x)+v(x)u^{'}(x)\)
explicitly convert into the core Integration by Parts formula?

Answer Review

1 How does the fundamental differentiation equation
\(\frac{d}{dx}[u(x)v(x)]=u(x)v^{'}(x)+v(x)u^{'}(x)\)
explicitly convert into the core Integration by Parts formula?

A By dividing both sides by the arbitrary constant \(C\).

B By integrating both sides with respect to \(x\) and isolating the integral of \(u(x)v^{'}(x)\).

C By taking the second derivative of the entire equation.

D By substituting \(u(x)=v(x)\)and eliminating the variables entirely.

💡 Short Explanation

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

✅ Correct Answer Explanation

The Integration by Parts formula is obtained by integrating the Product Rule and rearranging: \(\int u dv=uv-\int v du.\)

❓ Why Other Options Are Incorrect

Option A: The arbitrary constant is unrelated.
Option C: Second differentiation is unnecessary.
Option D: Substituting \(u=v\) does not derive the formula.

🎯 Strategy

Used

Elimination

Application: Identify the mathematical operation that reverses differentiation.

Final Logic: Integration of the Product Rule gives Integration by Parts.

🧠 Memory Trick

Product Rule → Integrate → Parts

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

2 Match the specific integrand choices in List I to their correctly determined evaluation format steps defined in List II.

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

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

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

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

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

💡 Short Explanation

Identify IBP structure Identify log structure Match standard reduction forms

✅ Correct Answer Explanation

1: x e^(2x) → IBP (x first, exponential second) → (d) 2: log x → log x first, 1 second → (c) 3: x(log x)² → (log x)² first → (b) 4: exponential-polynomial reduction → (a)

❓ Why Other Options Are Incorrect

Option B → swaps IBP structure incorrectly
Option C → mismatches log-based ordering
Option D → incorrect mapping of exponential case

🎯 Strategy

Used: Option Grouping

Final Logic: Each integrand matches unique IBP structure

🧠 Memory Trick

"Log first, exponential last"

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

3 When correctly evaluating
\(\int x^{2}log⁡x dx\)
using Integration by Parts, which of the following analytical choices and statements are entirely accurate?
A. \(\log\,x\) must mathematically be taken as the first function \(f(x)\).
B. \(x^{2}\)must be taken as the second function \(g(x)\).
C. The integral of the second function used in the formula will be \(\frac{x^{3}}{3}\).
D. The exponential function \(e^{x}\)should ideally be added to the equation.

A A, B

B B, C, D

C A, B, C

D A, D

💡 Short Explanation

Follow ILATE. Integrate the second function. Do not introduce extra functions.

✅ Correct Answer Explanation

Statements A, B and C are correct. Statement D is incorrect because \(e^{x}\)has no role in this integral.

❓ Why Other Options Are Incorrect

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

🎯 Strategy

Used

Elimination

Application: Check each statement independently.

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

🧠 Memory Trick

ILATE → Log First

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

4 Which of the following theoretical statements regarding the strict application of the ILATE rule is computationally INCORRECT?

A The ILATE rule is physically impossible to apply to integrands without an exponential term.

B The rule is a practical operational guide that dictates the first function \(f(x)\)based on the order of appearance in the acronym.

C If an integrand has an algebraic term and a logarithmic term, the algebraic term is chosen as the second function.

D It strictly mandates that an exponential function must always be integrated last.

💡 Short Explanation

ILATE is only a guideline. Exponential terms are not compulsory. Hence the statement is false.

✅ Correct Answer Explanation

The ILATE rule can be applied whenever appropriate, even if an exponential function is absent.

❓ Why Other Options Are Incorrect

Option D: Consistent with ILATE priority.
Option B: Correct description of the guideline.
Option C: Correct because logarithmic functions are preferred over algebraic functions.

🎯 Strategy

Used

Extreme Word Filter

Application: Look for absolute statements.

Final Logic: "Physically impossible" makes the statement incorrect.

🧠 Memory Trick

ILATE is a Guide, not a Rule

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

5 Execute the mixed formula rule for
\(\int e^{x}[f(x)+f^{'}(x)] dx\)
to perfectly evaluate
\(\int e^{x}\left(\frac{1}{x},\ \frac{1}{x^{2}}\right) dx.\)

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

B \(xe^{x}+C\)

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

D \(e^{x}log⁡∣x∣+C\)

💡 Short Explanation

Identify \(f(x)\). Apply the standard identity. Write the result.

✅ Correct Answer Explanation

Here, \(f(x)=\frac{1}{x},f^{'}(x)=-\frac{1}{x^{2}}.\) Using \(\int e^{x}[f(x)+f^{'}(x)] dx=e^{x}f(x)+C,\) we get \(e^{x}\left(\frac{1}{x}\right)+C.\)

❓ Why Other Options Are Incorrect

Option A: Uses only \(f^{'}(x)\).
Option B: Incorrect application.
Option D: Incorrect function identified.

🎯 Strategy

Used

Substitution

Application: Recognize the standard identity directly.

Final Logic: Substitute the correct \(f(x)\).

🧠 Memory Trick

\(e^{x}(f+f^{'})\rightarrow e^{x}f\)

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

6 Calculate the evaluated explicit result for the unconstrained polynomial–exponential integral:
\(\int xe^{x} dx.\)

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

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

C \(xe^{x}-e^{x}+C\)

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

💡 Short Explanation

Apply Integration by Parts. Take \(u=x\). Integrate \(e^{x}\).

✅ Correct Answer Explanation

Let \(u=x,dv=e^{x} dx.\) Then, \(du=dx,v=e^{x}.\) Hence, \(\int xe^{x} dx=xe^{x}-\int e^{x} dx=xe^{x}-e^{x}+C.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect sign arrangement.
Option B: Wrong application of Integration by Parts.
Option D: Incorrect antiderivative.

🎯 Strategy

Used

Substitution

Application: Differentiate the algebraic function and integrate the exponential function.

Final Logic: Apply the Integration by Parts formula directly.

🧠 Memory Trick

Algebraic First, Exponential Second

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

7 By manipulating terms and using Integration by Parts, accurately evaluate
\(\int \frac{\log\,x}{x^{2}} dx.\)
(Hint: Let \(\log\,x\) be the first function.)

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

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

C \(xlog⁡x+\frac{1}{x}+C\)

D \(-\frac{\log\,x}{x}-\frac{1}{x}+C\)

💡 Short Explanation

Apply Integration by Parts. Integrate \(x^{-2}\). Simplify the result.

✅ Correct Answer Explanation

Let \(u=log⁡x,dv=\frac{dx}{x^{2}}.\) Then, \(du=\frac{dx}{x},v=-\frac{1}{x}.\) Therefore, \(\int \frac{\log\,x}{x^{2}} dx=-\frac{\log\,x}{x}-\frac{1}{x}+C.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect sign.
Option B: Incorrect integration.
Option C: Unrelated expression.

🎯 Strategy

Used

Substitution

Application: Apply Integration by Parts using the ILATE rule.

Final Logic: Differentiate \(\log\,x\) and integrate \(x^{-2}\).

🧠 Memory Trick

Log → Parts

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

8 Evaluate the specific continuous moving parameter integral format natively expressed as: \(\int (log⁡x)^{2} dx.\)

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

B \(x(log⁡x)^{2}-2x+C\)

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

D \(x(log⁡x)^{2}-2xlog⁡x+2x+C\)

💡 Short Explanation

Apply Integration by Parts twice. Differentiate the logarithmic term. Simplify.

✅ Correct Answer Explanation

Using repeated Integration by Parts, \(\int (log⁡x)^{2}dx=x(log⁡x)^{2}-2xlog⁡x+2x+C.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect formula.
Option B: Missing the middle term.
Option C: Incorrect antiderivative.

🎯 Strategy

Used

Substitution

Application: Use repeated Integration by Parts.

Final Logic: Reduce the logarithmic power step by step.

🧠 Memory Trick

Log Power ↓ by Parts

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

9 Based on the ILATE rule hierarchy, if an integral contains both an inverse trigonometric function (I) and an algebraic function (A), which represents the correct designated choice for the first function?

A The algebraic function

B A completely arbitrary choice

C The inverse trigonometric function

D \(1\)

💡 Short Explanation

Apply ILATE priority. Inverse trigonometric functions come first. Algebraic functions come later.

✅ Correct Answer Explanation

According to the ILATE rule, inverse trigonometric functions have higher priority than algebraic functions. Therefore, they are chosen as the first function.

❓ Why Other Options Are Incorrect

Option A: Lower priority in ILATE.
Option B: Selection is not arbitrary.
Option D: Not chosen as the first function.

🎯 Strategy

Used

Option Grouping

Application: Recall the ILATE order.

Final Logic: I precedes A in ILATE.

🧠 Memory Trick

I before A

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

10 Systematically evaluate the nested, repeated Integration by Parts expression
\(\int x^{2}e^{x} dx.\)

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

B \(x^{3}e^{x}/3+C\)

C \(xe^{x}-e^{x}+C\)

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

💡 Short Explanation

Apply Integration by Parts twice. Reduce the polynomial degree. Simplify.

✅ Correct Answer Explanation

Repeated Integration by Parts gives \(\int x^{2}e^{x} dx=x^{2}e^{x}-2xe^{x}+2e^{x}+C.\)

❓ Why Other Options Are Incorrect

Option A: Missing the middle term.
Option B: Incorrect integration.
Option C: Corresponds to \(\int xe^{x} dx\).

🎯 Strategy

Used

Substitution

Application: Reduce the polynomial degree with each application.

Final Logic: Apply Integration by Parts until the polynomial becomes constant.

🧠 Memory Trick

Degree 2 → Parts Twice

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

11 Applying properties of bounds, explicitly calculate the numerical value of the definite integral
\(\int_{1}^{4}\,\frac{x}{\left(x+1)(x+4\right)} dx.\)

A \(log⁡ \left(\frac{8}{5}\right)-log⁡ \left(\frac{5}{2}\right)\)

B \(1\)

C \(\frac{4}{3}log⁡ \left(\frac{8}{5}\right)-\frac{1}{3}log⁡ \left(\frac{5}{2}\right)\)

D \(\frac{1}{3}log⁡ \left(\frac{8}{5}\right)+\frac{4}{3}log⁡ \left(\frac{5}{2}\right)\)

💡 Short Explanation

Use partial fractions. Integrate each term. Apply the limits.

✅ Correct Answer Explanation

Decomposing the integrand into partial fractions and evaluating between the limits \(1\) and \(4\) gives \(\frac{4}{3}log⁡ \left(\frac{8}{5}\right)-\frac{1}{3}log⁡ \left(\frac{5}{2}\right).\)

❓ Why Other Options Are Incorrect

Option A: Incorrect coefficients.
Option B: Incorrect numerical value.
Option D: Incorrect signs and coefficients.

🎯 Strategy

Used

Substitution

Application: Evaluate the antiderivative at the given limits.

Final Logic: Partial fractions followed by definite integration gives the required value.

🧠 Memory Trick

Split → Integrate → Apply Limits

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

12 According to standard boundary properties, what mathematical transformation correctly occurs to the fixed value of the integral if the lower and upper limits are identically swapped (i.e., changing \(\int_{a}^{b}\,a\textasciicircum b\) to \(\int_{b}^{a}\,b\textasciicircum a\))?

A The fixed value strictly becomes zero.

B The definite integral immediately reverts to an indefinite family of curves.

C The exact fixed value of the integral is multiplied by \(-1\).

D The upper limit mathematically doubles in value.

💡 Short Explanation

Interchanging limits changes the sign. Magnitude remains the same. Standard property of definite integrals.

✅ Correct Answer Explanation

The property \(\int_{a}^{b}\,f(x) dx=-\int_{b}^{a}\,f(x) dx\) shows that reversing the limits changes only the sign.

❓ Why Other Options Are Incorrect

Option A: The value does not become zero.
Option B: It remains a definite integral.
Option D: Limits do not double.

🎯 Strategy

Used

Elimination

Application: Recall the limit-reversal property.

Final Logic: Reversing limits changes only the sign.

🧠 Memory Trick

Reverse Limits → Reverse Sign

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

13 If an Area function is technically defined as
\(A(x)=\int_{0}^{x}\,3t^{2} dt,\)
what evaluates as the explicit derivative \(A^{'}(x)\)?

A \(6x\)

B \(3t^{2}+C\)

C \(3x^{2}\)

D \(x^{3}\)

💡 Short Explanation

Apply the First Fundamental Theorem. Differentiate the Area function. Replace \(t\) with \(x\).

✅ Correct Answer Explanation

By the First Fundamental Theorem, \(A^{'}(x)=3x^{2}.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect derivative.
Option B: Variable and constant are incorrect.
Option D: This is the antiderivative.

🎯 Strategy

Used

Substitution

Application: Apply the theorem directly.

Final Logic: Derivative of the Area function equals the integrand.

🧠 Memory Trick

Area′ = Integrand

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: First Fundamental Theorem of Calculus

14 Systematically apply the rigorous \(F(b)-F(a)\)evaluation to compute
\(\int_{0}^{3}\,x^{3} dx.\)

A \(9\)

B \(\frac{81}{4}\)

C \(\frac{27}{3}\)

D \(81\)

💡 Short Explanation

Find the antiderivative. Apply the limits. Simplify.

✅ Correct Answer Explanation

Since \(\int x^{3}dx=\frac{x^{4}}{4},{\left[\frac{x^{4}}{4}\right]}_{0}^{3}=\frac{81}{4}.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect evaluation.
Option C: Not obtained from the antiderivative.
Option D: Missing division by 4.

🎯 Strategy

Used

Substitution

Application: Evaluate the antiderivative at the limits.

Final Logic: Upper value minus lower value gives \(\frac{81}{4}\).

🧠 Memory Trick

Power +1, Divide

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Evaluation by Anti-derivatives

15 According to the bounded continuity properties of definite integrals, which mathematical equation successfully models splitting the integration interval \(\left[a,\ b\right]\)at point \(c\), where \(a<c<b\)?

A \(\int_{a}^{c}\,f(x) dx-\int_{c}^{b}\,f(x) dx\)

B \(\int_{a}^{b}\,f(x) dx=\int_{a}^{c}\,f(x) dx+\int_{c}^{b}\,f(x) dx\)

C \(\int_{a}^{c}\,f(x) dx\times \int_{c}^{b}\,f(x) dx\)

D \(\int_{a}^{b}\,f(x) dx=f(c)\)

💡 Short Explanation

Split the interval. Add the two integrals. Standard property.

✅ Correct Answer Explanation

A definite integral over \(\left[a,\ b\right]\)equals the sum of the integrals over \(\left[a,\ c\right]\)and \(\left[c,\ b\right]\).

❓ Why Other Options Are Incorrect

Option A: Uses subtraction.
Option C: Multiplication is incorrect.
Option D: Not a property of definite integrals.

🎯 Strategy

Used

Elimination

Application: Recall the interval-splitting property.

Final Logic: Definite integrals are additive over adjacent intervals.

🧠 Memory Trick

Split → Add

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

16 Calculate the precise numerical integration outcome for the definite integral
\(\int_{-1}^{1}\,x^{2}\sqrt{x^{3}+1} dx.\)

A \(\frac{4\sqrt{2}}{9}\)

B \(0\)

C \(\frac{2\sqrt{2}}{3}\)

D \(\frac{4}{9}\)

💡 Short Explanation

Use substitution. Change the limits. Evaluate the definite integral.

✅ Correct Answer Explanation

Let \(t=x^{3}+1,dt=3x^{2} dx.\) The limits become \(0\) and \(2\). Evaluating the transformed integral gives \(\frac{4\sqrt{2}}{9}.\)

❓ Why Other Options Are Incorrect

Option B: The integrand is not an odd function.
Option C: Incorrect numerical value.
Option D: Ignores the square-root term.

🎯 Strategy

Used

Substitution

Application: Convert the integral into a standard power integral.

Final Logic: Substitute first, then change the limits.

🧠 Memory Trick

\(x^{3}+1\rightarrow t\)

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

📄 Passage

In business economics, understanding market equilibrium is mathematically essential for determining the theoretical financial gains of consumers and producers. The Consumers' Surplus (CS) is explicitly calculated using definite integrals. When a market reaches an optimal equilibrium price p₀ and an equilibrium quantity x₀, there are invariably still buyers who would have been completely willing to pay a market price much higher than p₀. These specific buyers gain economically from the fact that the prevailing market price is only p₀. This overall economic gain is called the Consumers' Surplus. Geometrically, it is physically represented on a graph by the absolute area strictly below the continuous demand curve p = f(x) and strictly above the horizontal price line p = p₀. Mathematically,

Based on the analytical passage provided, what precise formula represents the Consumers' Surplus (CS)?

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

B \(CS=p_{0}x_{0}-\int_{0}^{x_{0}}\,f(x) dx\)

C \(CS=p_{0}x_{0}\int_{0}^{x_{0}}\,f(x) dx\)

D \(CS=\int_{0}^{x_{0}}\,f(x) dx+p_{0}x_{0}\)

💡 Short Explanation

Area under demand curve. Subtract the revenue rectangle. Gives Consumers' Surplus.

✅ Correct Answer Explanation

Consumers' Surplus equals the area under the demand curve from \(0\) to \(x_{0}\)minus the rectangle \(p_{0}x_{0}\).

❓ Why Other Options Are Incorrect

Option B: Formula is reversed.
Option C: Incorrect multiplication.
Option D: Rectangle should be subtracted, not added.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Match the formula directly with the passage.

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

🧠 Memory Trick

Area − Rectangle = CS

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

📄 Passage

If a market's demand curve is
\(p=25-x-x^{2},\)
and the equilibrium price is \(p_{0}=19\)(so that the equilibrium quantity is \(x_{0}=2\)), what is the definite integral representing the area under the demand curve before subtracting the rectangle?

A \(\int_{0}^{19}\,(25-x-x^{2}) dx\)

B \(19\times 2\)

C \(\int_{0}^{2}\,19 dx\)

D \(\int_{0}^{2}\,(25-x-x^{2}) dx\)

💡 Short Explanation

Integrate the demand function. Use equilibrium quantity. Rectangle is not included.

✅ Correct Answer Explanation

The required area is obtained by integrating the demand function from \(0\) to the equilibrium quantity \(2\).

❓ Why Other Options Are Incorrect

Option A: Uses the price as the upper limit.
Option B: Represents the rectangle area.
Option C: Integrates the price line instead of the demand curve.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Identify the expression described in the passage.

Final Logic: Area under the demand curve is the definite integral of the demand function.

🧠 Memory Trick

Demand Curve ⇒ Integrate

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

19 Compute the exact final evaluated area for the complex standard root expression\(\int_{0}^{2}\,\sqrt{x^{2}+4} dx.\)

A \(2\sqrt{2}+2log⁡(1+\sqrt{2})\)

B \(2\sqrt{2}-2log⁡(1+\sqrt{2})\)

C \(4log⁡(2+\sqrt{8})\)

D \(\sqrt{8}\)

💡 Short Explanation

Use the standard integral. Apply the limits. Simplify.

✅ Correct Answer Explanation

Using the standard formula for \(\int \sqrt{x^{2}+a^{2}} dx,\) and evaluating from \(0\) to \(2\), \(2\sqrt{2}+2log⁡(1+\sqrt{2}).\)

❓ Why Other Options Are Incorrect

Option B: Incorrect sign.
Option C: Incorrect expression.
Option D: Incomplete evaluation.

🎯 Strategy

Used

Substitution

Application: Apply the standard integral formula.

Final Logic: Evaluate the antiderivative at the limits.

🧠 Memory Trick

Root Form → Standard Formula

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

20 Accurately transform the limits and evaluate the bounded substitution matrix specifically written as
\(\int_{-3}^{0}\,x\sqrt{x+4} dx.\)

A \(-\frac{116}{15}\)

B \(\frac{116}{15}\)

C \(\frac{94}{15}\)

D \(-\frac{94}{15}\)

💡 Short Explanation

Apply substitution. Change the limits. Evaluate the transformed integral.

✅ Correct Answer Explanation

Let \(t=x+4.\) The limits become \(1\) and \(4\). Evaluating the transformed integral gives \(-\frac{116}{15}.\)

❓ Why Other Options Are Incorrect

Option B: Incorrect sign.
Option C: Incorrect numerical value.
Option D: Incorrect numerical value.

🎯 Strategy

Used

Substitution

Application: Substitute the variable and update the limits.

Final Logic: Correct substitution with transformed limits gives the required result.

🧠 Memory Trick

New Variable → New Limits

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

CUET UG Applied Mathematics Booster Test 2 - Integration by Parts and Definite Integrals

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