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Integrals Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Integrals Set-1

Practice Important MCQs for Integrals — Mathematics, Class 12.

Integrals are a core concept in Class 12 Mathematics because they connect areas under curves with differential changes and lead to powerful techniques like substitution, symmetry, and handling improper integrals. In board exams, they test your ability to evaluate definite/infinite integrals correctly and efficiently, while in competitive exams they also test deeper skills such as parameter differentiation, logarithmic integral identities, and convergence analysis.

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Q1. Evaluate the definite integral

I=\int_{0}^{\pi} x\sin x\,dx.

(A)

$0$

(B)

$2$

(C)

$\pi$

(D)

$1$

Q2. Compute the integral

I=\int_{0}^{\pi/4}\ln\big(1+\tan x\big)\,dx.

(A)

$\dfrac{\pi}{8}\ln 2$

(B)

$\dfrac{\pi}{4}\ln 2$

(C)

$\dfrac{\ln 2}{2}$

(D)

$0$

Q3. Evaluate

I=\int_{0}^{1}\frac{\arctan x}{1+x^{2}}\,dx.

(A)

$\dfrac{\pi^{2}}{16}$

(B)

$\dfrac{\pi^{2}}{8}$

(C)

$\dfrac{\pi^{2}}{24}$

(D)

$\dfrac{\pi^{2}}{32}$

Q4. Assertion (A): The improper integral

\int_{0}^{\infty}\frac{\sin x}{x}\,dx

converges.
Reason (R): The integral

\int_{0}^{\infty}\left|\frac{\sin x}{x}\right|\,dx

converges.

(A)

Both (A) and (R) are true and (R) is a correct explanation of (A).

(B)

Both (A) and (R) are true but (R) is not a correct explanation of (A).

(C)

(A) is true but (R) is false.

(D)

(A) is false but (R) is true.

Q5. Evaluate

I=\int_{0}^{\pi}\frac{x}{1+\sin^{2}x}\,dx.

(A)

$\dfrac{\pi^{2}}{4}$

(B)

$\dfrac{\pi^{2}}{2\sqrt{2}}$

(C)

$\dfrac{\pi^{2}}{2}$

(D)

$\dfrac{\pi^{2}}{8}$

Q6. Evaluate the integral $I=\displaystyle\int_{0}^{\pi} x\sin x\,dx$ .

(A)

$2$

(B)

$\dfrac{\pi}{2}$

(C)

$\pi$

(D)

$2\pi$

Q7. Evaluate the definite integral $I=\displaystyle\int_{0}^{\pi}\frac{x\sin x}{1+\cos^{2}x}\,dx$ .

(A)

$\dfrac{\pi^{2}}{4}$

(B)

$\dfrac{\pi}{2}$

(C)

$\dfrac{\pi^{2}}{8}$

(D)

$0$

Q8. Compute the improper integral $I=\displaystyle\int_{0}^{\infty}\frac{\arctan x}{1+x^{2}}\,dx$ .

(A)

$\dfrac{\pi^{2}}{12}$

(B)

$\dfrac{\pi}{2}$

(C)

$\dfrac{\pi^{2}}{4}$

(D)

$\dfrac{\pi^{2}}{8}$

Q9. Let $I(a)=\displaystyle\int_{0}^{\pi}\ln\big(1-2a\cos x+a^{2}\big)\,dx$ for real $a$ . Which of the following gives the correct value of $I(a)$ ?

(A)

$I(a)=\pi\ln(1+a^{2})\quad\text{for all real }a$

(B)

$I(a)=\begin{cases}0,&|a|\le 1,\$$4pt]2\pi\ln|a|,&|a|>1\end{cases}$

(C)

$I(a)=2\pi\ln(1+a)\quad\text{for }a>-1$

(D)

$I(a)=0\quad\text{for all real }a$

Q10. For real $p>-1$ evaluate $I(p)=\displaystyle\int_{0}^{1}\frac{x^{p}-1}{\ln x}\,dx$ .

(A)

$\dfrac{1}{p+1}$

(B)

$\dfrac{p}{p+1}$

(C)

$\ln(p+1)$

(D)

$-\dfrac{1}{(p+1)^{2}}$

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