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Integrals Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Integrals Set-4

Practice Important MCQs for Integrals — Mathematics, Class 12.

Integrals form a core backbone of Class 12 Mathematics and are heavily used in board exams as well as competitive exams. Mastery of definite integrals, techniques like substitution and symmetry, and special results such as integrals involving $\ln x$ (parameter differentiation) helps you solve a wide range of “application-based” and “formula-based” questions quickly and accurately.

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Q1. Evaluate the definite integral $\displaystyle \int_{0}^{1}\frac{2x}{(1+x^{2})^{2}}\,dx$ .

(A)

$\dfrac{1}{4}$

(B)

$\dfrac{3}{8}$

(C)

$\dfrac{1}{2}$

(D)

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

Q2. Evaluate the integral $\displaystyle \int_{0}^{1}\frac{\tan^{-1}x}{1+x^{2}}\,dx$ .

(A)

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

(B)

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

(C)

$\dfrac{\pi}{8}$

(D)

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

Q3. Find the value of $\displaystyle \int_{0}^{\pi}\frac{x\sin x}{1+\cos^{2}x}\,dx$ .

(A)

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

(B)

$\dfrac{\pi^{2}}{6}$

(C)

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

(D)

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

Q4. For real numbers $m,n>-1$ , evaluate $\displaystyle \int_{0}^{1}\frac{x^{m}-x^{n}}{\ln x}\,dx$ .

(A)

$\dfrac{1}{m+1}-\dfrac{1}{n+1}$

(B)

$\ln\!\left(\dfrac{m+1}{\,n+1\,}\right)$

(C)

$\dfrac{m-n}{(m+1)(n+1)}$

(D)

$\ln\!\left(\dfrac{n+1}{\,m+1\,}\right)$

Q5. Evaluate the integral $\displaystyle \int_{0}^{1}\frac{\ln(1+x)}{x}\,dx$ .

(A)

$\dfrac{\pi^{2}}{6}$

(B)

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

(C)

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

(D)

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

Q6. Evaluate the definite integral

\int_0^1 \frac{x}{(1+x^2)^2}\,dx

(A)

$\dfrac{3}{8}$

(B)

$\dfrac{1}{8}$

(C)

$\dfrac{1}{4}$

(D)

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

Q7. Evaluate

\int_0^{\pi/2} \sin^3 x\,\cos^2 x\,dx

(A)

$\dfrac{2}{15}$

(B)

$\dfrac{1}{6}$

(C)

$\dfrac{1}{10}$

(D)

$\dfrac{7}{45}$

Q8. Evaluate

\int_0^1 \frac{\ln(1+x)}{x}\,dx

(A)

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

(B)

$\dfrac{\pi^2}{6}$

(C)

$\ln 2$

(D)

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

Q9. Evaluate

\int_0^1 \frac{x^2 - x^5}{\ln x}\,dx

(A)

$\ln\!\dfrac{3}{2}$

(B)

$-\ln 2$

(C)

$\ln\!\dfrac{1}{3}$

(D)

$0$

Q10. Evaluate

\int_0^{\pi/4} \frac{dx}{\cos^2 x + \sin^2 x + \sin 2x}

(A)

$\dfrac{1}{2}$

(B)

$1$

(C)

$\dfrac{1}{4}$

(D)

$\dfrac{\sqrt{2}}{2}$

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

(A)

$\dfrac{1}{2}$

(B)

$\dfrac{1}{3}$

(C)

$\dfrac{\pi}{4}$

(D)

$\dfrac{\pi}{6}$

Q12. Compute the integral $I=\int_{0}^{\pi}\frac{x\sin x}{1+\cos^{2}x}\,dx$ .

(A)

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

(B)

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

(C)

$0$

(D)

$\dfrac{\pi}{2}$

Q13. Evaluate $I=\int_{0}^{1}\frac{\arctan x}{1+x^{2}}\,dx$ .

(A)

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

(B)

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

(C)

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

(D)

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

Q14. For real $a>-1$ , evaluate $I(a)=\displaystyle\int_{0}^{1}\frac{x^{a}-1}{\ln x}\,dx$ .

(A)

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

(B)

$\ln(a+1)$

(C)

$\ln a \quad\text{(for }a>0\text{)}$

(D)

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

Q15. Find the value of the integral $I=\displaystyle\int_{0}^{1}\frac{\ln(1+x)}{x}\,dx$ .

(A)

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

(B)

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

(C)

$\dfrac{\pi^{2}}{6}$

(D)

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

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