Section A — MCQ (Single Correct)
Question 1
Evaluate the following trigonometric integral expression: $\int \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \, dx$.
A
$\sec x - \csc x + C$
B
$\sec x + \csc x + C$
C
$-\sec x - \csc x + C$
D
$\tan x - \cot x + C$
Question 2
Solve the substitution integral $\int \frac{e^x(1+x)}{\cos^2(x e^x)} \, dx$:
A
$\tan(x e^x) + C$
B
$\cot(x e^x) + C$
C
$\sec(x e^x) + C$
D
$-\tan(x e^x) + C$
Question 3
Find the value of the algebraic radical standard integral: $\int \frac{1}{\sqrt{x^2 + 4x + 1}} \, dx$.
A
$\ln\left|(x+2) + \sqrt{x^2+4x+1}\right| + C$
B
$\ln\left|(x+2) + \sqrt{x^2+4x+4}\right| + C$
C
$\sin^{-1}\left(\frac{x+2}{\sqrt{3}}\right) + C$
D
$\frac{1}{2}\ln\left|(x+2) + \sqrt{x^2+4x+1}\right| + C$
Question 4
Evaluate the exponential product form using Integration by Parts: $\int x^2 e^{2x} \, dx$.
A
$\frac{1}{2}x^2 e^{2x} - \frac{1}{2}x e^{2x} + \frac{1}{4}e^{2x} + C$
B
$\frac{1}{2}x^2 e^{2x} + \frac{1}{2}x e^{2x} - \frac{1}{4}e^{2x} + C$
C
$x^2 e^{2x} - 2x e^{2x} + 2e^{2x} + C$
D
$\frac{1}{2}x^2 e^{2x} - \frac{1}{4}x e^{2x} + C$
Question 5
The value of the integral $\int \frac{1}{x(x^5 + 1)} \, dx$ evaluated via substitution is:
A
$\frac{1}{5}\ln\left|\frac{x^5}{x^5+1}\right| + C$
B
$\frac{1}{5}\ln\left|\frac{x^5+1}{x^5}\right| + C$
C
$\ln\left|\frac{x^5}{x^5+1}\right| + C$
D
$\frac{1}{5}\tan^{-1}(x^5) + C$
Question 6
Solve the exponential structural shortcut form: $\int e^x \left( \ln x + \frac{1}{x} \right) \, dx$.
A
$e^x \ln x + C$
B
$\frac{e^x}{x} + C$
C
$-e^x \ln x + C$
D
$e^x \left(\ln x - \frac{1}{x}\right) + C$
Question 7
Evaluate the definite radical form $\int \sqrt{a^2 - x^2} \, dx$ using trigonometric substitution:
A
$\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$
B
$x\sqrt{a^2-x^2} + a^2\sin^{-1}\left(\frac{x}{a}\right) + C$
C
$\frac{x}{2}\sqrt{a^2-x^2} - \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$
D
$\frac{1}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$
Question 8
The integration of the rational function $\int \frac{3x+1}{(x-1)(x-2)} \, dx$ using partial fractions yields:
A
$7\ln|x-2| - 4\ln|x-1| + C$
B
$7\ln|x-1| - 4\ln|x-2| + C$
C
$4\ln|x-2| - 7\ln|x-1| + C$
D
$\ln|(x-1)(x-2)| + C$
Question 9
Find the value of the algebraic product integral: $\int x \ln(x+1) \, dx$.
A
$\frac{1}{2}(x^2-1)\ln(x+1) - \frac{1}{4}x^2 + \frac{1}{2}x + C$
B
$\frac{1}{2}x^2\ln(x+1) - \frac{1}{4}x^2 + C$
C
$\frac{1}{2}(x^2-1)\ln(x+1) + \frac{1}{4}x^2 - \frac{1}{2}x + C$
D
$\frac{1}{2}x^2\ln(x+1) - \frac{1}{2}x + C$
Question 10
Evaluate the inverse trigonometric form $\int \sin^{-1} x \, dx$ using IBP:
A
$x\sin^{-1} x + \sqrt{1-x^2} + C$
B
$x\sin^{-1} x - \sqrt{1-x^2} + C$
C
$\frac{1}{\sqrt{1-x^2}} + C$
D
$x\sin^{-1} x + C$
Section B — Integer Type
Question 11 — Integer answer
If $\int \frac{1}{x^2 + 6x + 25} \, dx = \frac{1}{k}\tan^{-1}\left(\frac{x+3}{k}\right) + C$, find the value of the integer parameter $k$.
Question 12 — Integer answer
Find the value of the leading constant coefficient $A$ if the partial fraction decomposition satisfies $\frac{1}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1}$.
Question 13 — Integer answer
If the tangent reduction formula satisfies $\int \tan^5 x \, dx = \frac{1}{k}\tan^4 x - \int \tan^3 x \, dx$, find the value of the integer denominator $k$.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): The family of anti-derivatives for the function $f(x) = \frac{1}{x}$ is uniquely given by $F(x) = \ln x + C$ across its complete real definition layout.Reason (R): The natural logarithmic absolute map derivative satisfies $\frac{d}{dx}(\ln|x|) = \frac{1}{x}$ for all non-zero real inputs.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: A is false (because the domain includes negative numbers, requiring the absolute value $.
Question 15 — Assertion / Reason
Assertion (A): The exponential identity form $\int e^x [f(x) + f'(x)] \, dx$ resolves directly to $e^x f(x) + C$.Reason (R): This structural shortcut is derived by separating the terms and applying Integration by Parts to the $\int e^x f(x) \, dx$ component.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: Both A and R are true, and R is the correct explanation.