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Integrals Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Integrals Set-3

Practice Important MCQs for Integrals — Mathematics, Class 12.

Integrals are central to Class 12 Mathematics and frequently appear in board exams and competitive tests because they model real quantities like area, volume, displacement, and accumulated change. Strong command of techniques such as substitution, symmetry, partial fractions, and parameter differentiation helps students evaluate definite integrals accurately and also builds the foundation needed for higher-level calculus problems in JEE/NEET.

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Q1. Evaluate

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

(A)

$\ln 2$

(B)

$\dfrac{1}{2}\ln 2$

(C)

$\dfrac{1}{4}(\ln 2)^{2}$

(D)

$\dfrac{1}{2}$

Q2. Let

I(a)=\int_{0}^{1}\frac{x^{a}-1}{\ln x}\,dx,\qquad a>-1.

Find $I(a)$ .

(A)

$-\ln(a+1)$

(B)

$\dfrac{1}{a+1}$

(C)

$2\ln(a+1)$

(D)

$\ln(a+1)$

Q3. Evaluate

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

(A)

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

(B)

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

(C)

$\dfrac{\pi^{2}}{64}$

(D)

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

Q4. For $|a|<1$ , evaluate

\int_{0}^{\pi}\ln\bigl(1+a\cos x\bigr)\,dx.

(A)

$0$

(B)

$\pi\ln\!\bigl(1+\tfrac{a}{2}\bigr)$

(C)

$\displaystyle \pi\ln\!\Big(\frac{1+\sqrt{1-a^{2}}}{2}\Big)$

(D)

$\dfrac{\pi}{2}\ln\!\bigl(1-a^{2}\bigr)$

Q5. Evaluate

\int_{0}^{\infty}\frac{\ln(1+x^{2})}{1+x^{2}}\,dx.

(A)

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

(B)

$\pi\ln 2$

(C)

$2\ln 2$

(D)

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

Q6. Evaluate the integral $\displaystyle \int_{0}^{1}\frac{1-x}{1+x^{2}}\,dx$ .

(A)

$\displaystyle \frac{\pi}{4}+\frac{1}{2}\ln 2$

(B)

$\displaystyle \frac{\pi}{4}-\ln 2$

(C)

$\displaystyle \frac{\pi}{4}-\frac{1}{2}\ln 2$

(D)

$\displaystyle \frac{\pi}{4}-\frac{1}{4}\ln 2$

Q7. Let $I=\displaystyle \int_{0}^{\pi}\frac{x\sin x}{1+\cos^{2}x}\,dx$ . Compute $I$ .

(A)

$\displaystyle \frac{\pi^{2}}{4}$

(B)

$\displaystyle \frac{\pi^{2}}{2}$

(C)

$\displaystyle \frac{\pi}{4}$

(D)

$\displaystyle \pi\ln 2$

Q8. Evaluate the integral $\displaystyle \int_{0}^{1}\frac{\arctan x}{1+x^{2}}\,dx$ .

(A)

$\displaystyle \frac{\pi^{2}}{16}$

(B)

$\displaystyle \frac{\pi}{8}$

(C)

$\displaystyle \frac{\pi^{2}}{24}$

(D)

$\displaystyle \frac{\pi^{2}}{32}$

Q9. If $I=\displaystyle \int_{0}^{1}\frac{x^{2}-1}{\ln x}\,dx$ , find the value of $I$ .

(A)

$\displaystyle \ln 2$

(B)

$\displaystyle \ln 3$

(C)

$\displaystyle 2\ln 2$

(D)

$\displaystyle \ln\!\frac{3}{2}$

Q10. Evaluate the integral $\displaystyle \int_{0}^{2\pi}\frac{dx}{5+3\cos x}$ .

(A)

$\displaystyle \frac{\pi}{2}$

(B)

$\displaystyle \frac{\pi}{4}$

(C)

$\displaystyle \pi$

(D)

$\displaystyle \frac{2\pi}{5}$

Q11. Evaluate the integral $I=\displaystyle\int_{0}^{1}\dfrac{x^{2}}{\sqrt{1-x^{2}}}\,dx$ .

(A)

$\dfrac{1}{2}$

(B)

$\dfrac{\pi}{6}$

(C)

$\dfrac{\pi}{4}$

(D)

$1$

Q12. Evaluate the improper integral $I=\displaystyle\int_{0}^{\infty}\dfrac{x}{1+x^{4}}\,dx$ .

(A)

$\dfrac{\pi}{4}$

(B)

$\dfrac{\pi}{2}$

(C)

$\dfrac{\pi}{8}$

(D)

$\dfrac{1}{2}$

Q13. Let $a,b>-1$ with $a\ne b$ . Evaluate $I=\displaystyle\int_{0}^{1}\dfrac{x^{a}-x^{b}}{\ln x}\,dx$ .

(A)

$\dfrac{a-b}{a+b+2}$

(B)

$\ln\!\left(\dfrac{b+1}{a+1}\right)$

(C)

$\dfrac{\ln(a+1)-\ln(b+1)}{a-b}$

(D)

$\ln\!\left(\dfrac{a+1}{b+1}\right)$

Q14. For positive constants $a,b>0$ , evaluate

I=\int_{0}^{\infty}\dfrac{\cos(ax)-\cos(bx)}{x^{2}}\,dx.

(A)

$\dfrac{\pi}{2}\,(a-b)$

(B)

$\dfrac{\pi}{2}\,(b-a)$

(C)

$\pi\,(b-a)$

(D)

$\dfrac{b-a}{\pi}$

Q15. Evaluate $I=\displaystyle\int_{0}^{1}\dfrac{\arctan\sqrt{x}}{\sqrt{x}}\,dx$ .

(A)

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

(B)

$\dfrac{\pi}{4}-\dfrac{1}{2}\ln 2$

(C)

$\ln 2$

(D)

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

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