Definite Integrals Important Questions with Solutions (Q.132 – Q.252)

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Definite Integrals Important Questions with Solutions (Q.132 – Q.252)

In this post, you will find Definite Integrals Important Questions (Q.132 – Q.252) with four options and correct answers. This set is very useful for JEE Mains, JEE Advanced, NEET, UP TGT PGT, and other competitive exams. The content is WordPress friendly, mobile responsive, and SEO optimized for quick reading and revision.

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Table of Contents

Introduction to Definite Integrals

Definite integrals are one of the most important topics in Calculus. These questions frequently appear in JEE, NEET, and TGT/PGT Mathematics exams. Below we provide a detailed set of solved questions ranging from Q.132 to Q.252.

Definite Integrals Questions (132 – 150)

Here is the list of important definite integrals questions from 132 to 150 with answers.

Definite Integrals Questions (151 – 180)

Below you will find questions 151 to 180 with multiple-choice answers and solutions.

Definite Integrals Questions (181 – 210)

This section covers definite integrals problems from Q.181 to Q.210.

Definite Integrals Questions (211 – 230)

Practice set of questions between 211 to 230 with correct answers.

Definite Integrals Questions (231 – 252)

Final set of definite integrals questions from 231 to 252 with solutions.

Conclusion & Exam Tips

By practicing these Definite Integral MCQs (132–252), students can improve speed, accuracy, and understanding of calculus problems. For best results, revise formulas, practice previous year papers, and attempt online mock tests.

Definite Integrals Q&A (132-252)

Definite Integrals — Questions & Answers (Q.132 — Q.252)

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Q.132. If \(\dfrac{1}{\sqrt{\alpha}}\displaystyle\int_{1}^{\alpha}\Big(\tfrac{3}{2}\sqrt{x}+1-\tfrac{1}{\sqrt{x}}\Big)\,dx<4\), then \(\alpha\) may take the value :
(a) \(0\)
(b) \(4\)
(c) \(9\)
(d) \(13\sqrt{2}\)
Answer: (d) \(13\sqrt{2}\)
Q.133. If \(I_n=\displaystyle\int_{0}^{\infty} e^{-x}x^{\,n-1}\,dx\), then \(\displaystyle\int_{0}^{\infty} e^{-\lambda x} x^{\,n-1}\,dx\) equals :
(a) \(\lambda^{-(n-1)}I_n\)
(b) \(\lambda^{-1}I_n\)
(c) \(\lambda^{-n}I_n\)
(d) \(2\lambda^{-(n+1)}I_n\)
Answer: (c) \(\lambda^{-n}I_n\)
Q.134. Let \(F'(x)=\dfrac{e^{-\sin x}}{x}\) for \(x>0\) and \(\displaystyle\int_{1}^{4}\dfrac{2e^{\sin x^{2}}}{x}\,dx=F(k)-F(1)\). Then \(k=\) :
(a) \(16\)
(b) \(12\)
(c) \(9\)
(d) \(8\)
Answer: (c) \(9\)
Q.135. If \(\displaystyle\int_{0}^{\infty} e^{-\alpha x}\,dx=\dfrac{1}{\alpha}\), then evaluate \(\displaystyle\int_{0}^{\infty} x^{n} e^{-\alpha x}\,dx\).
(a) \(\dfrac{n!}{\alpha^{n}}\)
(b) \(\dfrac{n!}{\alpha^{n+1}}\)
(c) \(\dfrac{1}{\alpha^{n+1}}\)
(d) \((-1)^n\) something
Answer: (b) \(\dfrac{n!}{\alpha^{n+1}}\)
Q.136. If \(\displaystyle\int_{0}^{\infty} e^{-x^{2}}\,dx=\dfrac{\sqrt{\pi}}{2}\), then for \(a>0\), \(\displaystyle\int_{0}^{\infty} e^{-a x^{2}}\,dx=\) :
(a) \(\dfrac{\sqrt{\pi a}}{2}\)
(b) \(\dfrac{a\sqrt{\pi}}{2}\)
(c) \(\dfrac{\sqrt{\pi}}{2a}\)
(d) \(\dfrac{1}{2}\sqrt{\dfrac{\pi}{a}}\)
Answer: (d) \(\dfrac{1}{2}\sqrt{\dfrac{\pi}{a}}\)
Q.137. Given \(I_1=\int_{0}^{x} e^{-t}e^{-t^{2}}\,dt\) and \(I_2=\int_{0}^{x} \dfrac{e^{-t^{2}}}{4}\,dt\). Relation between \(I_1\) and \(I_2\) is :
(a) \(I_1=e^{2x^{2}}I_2\)
(b) \(I_1=e^{x^{2}}I_2\)
(c) \(I_1=e^{x^{2}/3}I_2\)
(d) \(I_1=e^{x^{2}/4}I_2\)
Answer: (c)
Q.138. If \(\alpha f(x)+\beta f(1/x)=1/x-5\) for \(x\ne0\) and \(\alpha\ne\beta\), then \(\displaystyle\int_{1}^{2} f(x)\,dx=\) :
(a) \(\dfrac{2\alpha\ln2-5\alpha+3\beta}{2(\alpha^{2}-\beta^{2})}\)
(b) \(\dfrac{3\alpha\ln2-7\alpha+5\beta}{2(\alpha^{2}-\beta^{2})}\)
(c) \(\dfrac{\alpha\ln2-5\alpha+3\beta}{2(\alpha^{2}-\beta^{2})}\)
(d) \(\dfrac{2\alpha\ln2-10\alpha+7\beta}{2(\alpha^{2}-\beta^{2})}\)
Answer: (c)
Q.139. Let \(a,b,c\) be non-zero reals with given integral condition (book). Then the quadratic \(ax^{2}+bx+c=0\) has :
(a) no root in \((0,2)\)
(b) at least one root in \((1,2)\)
(c) double root in \((0,2)\)
(d) two imaginary roots
Answer: (b)
Q.140. Let \(f'(x)=f(x)\), \(f(0)=1\) and \(f(x)+g(x)=2x\). Then \(\displaystyle\int_{0}^{1} f(x)g(x)\,dx=\) :
(a) \(-\tfrac{1}{2}(e^{2}-3)\)
(b) \(\tfrac{1}{2}(e-3)\)
(c) \(-\tfrac{1}{2}(e^{2}-5)\)
(d) \(\tfrac{1}{2}(e+5)\)
Answer: (c)
Q.141. If \((1+x)^{n}=C_{0}+C_{1}x+\cdots+C_{n}x^{n}\), value of \(C_{0}+\tfrac{1}{2}C_{1}+\tfrac{1}{3}C_{2}+\cdots+\tfrac{1}{n+1}C_{n}\) equals :
(a) \(\dfrac{2^{\,n-1}}{n+1}\)
(b) \(\dfrac{2^{\,n+1}}{n+1}\)
(c) \(\dfrac{2^{\,n-1}-1}{n+1}\)
(d) \(\dfrac{2^{\,n+1}-1}{n+1}\)
Answer: (d)
Q.142. Evaluate \(\displaystyle\int_{0}^{\pi}\dfrac{\sin\big((n+\tfrac12)x\big)}{\sin(x/2)}\,dx\) for integer \(n\).
(a) \(0\)
(b) \(\pi/4\)
(c) \(\pi/2\)
(d) \(\pi\)
Answer: (b)
Q.143. \(\displaystyle\int_{0}^{2a} f(x)\,dx – \int_{0}^{a} f(2a-x)\,dx =\) :
(a) \(\displaystyle\int_{0}^{a} f(x)\,dx\)
(b) \(\displaystyle\int_{0}^{2a} f(x)\,dx\)
(c) \(\displaystyle\int_{0}^{a} f(a+x)\,dx\)
(d) \(\displaystyle\int_{0}^{2a} f(a+x)\,dx\)
Answer: (d)
Q.144. Evaluate \(\displaystyle\int_{0}^{\pi/2}\dfrac{f(x)}{f(x)+f(\tfrac{\pi}{2}-x)}\,dx\).
(a) \(0\)
(b) \(\pi/4\)
(c) \(\pi/2\)
(d) \(\pi\)
Answer: (b) \(\pi/4\)
Q.145. Evaluate \(\displaystyle\int_{0}^{2a}\dfrac{f(x)}{f(x)+f(2a-x)}\,dx\).
(a) \(0\)
(b) \(a\)
(c) \(2a\)
(d) \(4a\)
Answer: (c) \(2a\)
Q.146. Evaluate \(\displaystyle\int_{0}^{a}\dfrac{f(x)}{f(x)+f(a-x)}\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(a/2\)
(d) \(a\)
Answer: (b) \(1\)
Q.147. Evaluate \(\displaystyle\int_{a}^{b}\dfrac{x f(x)}{f(x)+f(a+b-x)}\,dx\).
(a) \(\tfrac{1}{2}(b-a)\)
(b) \(2(a+b)\)
(c) \(\dfrac{b-a}{a+b}\)
(d) \(\dfrac{b^{2}-a^{2}}{2a}\)
Answer: (a) \(\tfrac{1}{2}(b-a)\)
Q.148. Evaluate \(\displaystyle\int_{0}^{a}[f(x)+f(-x)]\,dx\).
(a) \(0\)
(b) \(2\displaystyle\int_{0}^{a}f(x)\,dx\)
(c) \(\displaystyle\int_{a}^{0}f(x)\,dx\)
(d) \(-\displaystyle\int_{-a}^{a} f(-x)\,dx\)
Answer: (c)
Q.149. Evaluate \(\displaystyle\int_{0}^{a}[f(a-x)+f(a+x)]\,dx.\)
(a) \(0\)
(b) \(\displaystyle\int_{-a}^{a} f(x)\,dx\)
(c) \(2\displaystyle\int_{0}^{a} f(x)\,dx\)
(d) \(\displaystyle\int_{0}^{2a} f(x)\,dx\)
Answer: (d)
Q.150. If \(f(a-x)=f(x)\), then \(\displaystyle\int_{0}^{a} x f(x)\,dx =\) :
(a) \(2a^{2}\displaystyle\int_{0}^{a} f(x)\,dx\)
(b) \(a\displaystyle\int_{0}^{a} f(x)\,dx\)
(c) \(\dfrac{a}{2}\displaystyle\int_{0}^{a} f(x)\,dx\)
(d) \(\dfrac{a^{2}}{2}\displaystyle\int_{0}^{a} f(x)\,dx\)
Answer: (c) \(\dfrac{a}{2}\displaystyle\int_{0}^{a} f(x)\,dx\)
Q.151. If \(f(a+b-x)=f(x)\), then \(\displaystyle\int_{a}^{b} x f(x)\,dx =\) :
(a) \(0\)
(b) \(\dfrac{a+b}{2}\displaystyle\int_{a}^{b} f(x)\,dx\)
(c) \(\dfrac{b-a}{2}\displaystyle\int_{a}^{b} f(x)\,dx\)
(d) \(\dfrac{b^{2}-a^{2}}{2}\displaystyle\int_{a}^{b} f(x)\,dx\)
Answer: (d)
Q.152. Let \(f\) be positive. \(I_1=\displaystyle\int_{-k}^{k} x f(1-x)\,dx\) and \(I_2=\displaystyle\int_{-k}^{k} x f(1-x)\,dx\) where \(2k-1>0\). Then \(I_1/I_2=\) :
(a) \(2\)
(b) \(-1\)
(c) \(1/2\)
(d) \(1\)
Answer: (c)
Q.153. If \(f(x)=\dfrac{e^{x}}{1+e^{x}}\), \(I_1=\displaystyle\int_{f(a)}^{f(-a)} x\,g\{x(1-x)\}\,dx\) and \(I_2=\displaystyle\int_{f(a)}^{f(-a)} g\{x(1-x)\}\,dx\), then \(I_2/I_1=\) :
(a) \(1/2\)
(b) \(1\)
(c) \(3/2\)
(d) \(2\)
Answer: (b)
Q.154. \(\displaystyle\int_{0}^{1} x f(\sin x)\,dx =\) :
(a) \(0\)
(b) \(\pi/2\)
(c) \(\pi\displaystyle\int_{0}^{\pi} f(\sin x)\,dx\)
(d) \(\dfrac{\pi}{2}\displaystyle\int_{0}^{\pi} f(\sin x)\,dx\)
Answer: (d)
Q.155. If \(\displaystyle\int_{0}^{\pi} x f(\sin x)\,dx = A\), and \(\displaystyle\int_{0}^{\pi/2} f(\sin x)\,dx = ?\) then \(A=\) :
(a) \(0\)
(b) \(\pi/4\)
(c) \(\pi\)
(d) \(2\pi\)
Answer: (c)
Q.156. \(\displaystyle\int_{-\pi/2}^{\pi/2} f(\cos x)\,dx =\) :
(a) \(0\)
(b) \(\pi\)
(c) \(\displaystyle\int_{0}^{\pi} f(\cos x)\,dx\)
(d) \(2\displaystyle\int_{0}^{\pi/2} f(\sin x)\,dx\)
Answer: (c)
Q.157. The function \(\Phi(x)=\displaystyle\int_{a}^{x}\ln\big(t+\sqrt{1+t^{2}}\big)\,dt\) is :
(a) even
(b) odd
(c) periodic
(d) none of these
Answer: (a)
Q.158. If \(f(x)=\displaystyle\int_{0}^{x}\ln\Big(\dfrac{1-t}{1+t}\Big)\,dt\), then \(f(x)\) is :
(a) odd
(b) periodic
(c) symmetric
(d) even
Answer: (d)
Q.159. If \(f(t)\) is odd, then \(F(x)=\displaystyle\int_{a}^{x} f(t)\,dt\) is :
(a) odd
(b) even
(c) can be odd or even
(d) none
Answer: (b)
Q.160. Evaluate \(\displaystyle\int_{-\pi}^{\pi}\sin^{3}x\cos^{2}x\,dx =\) ?
(a) \(0\)
(b) \(1\)
(c) \(\pi\)
(d) \(2\pi\)
Answer: (c)
Q.161. Evaluate \(\displaystyle\int_{-\pi}^{\pi}(x+x^{3})\,dx =\) ?
(a) \(0\)
(b) \(\pi/2\)
(c) \(\pi\)
(d) \(2\pi\)
Answer: (a)
Q.162. Evaluate \(\displaystyle\int_{-\pi/4}^{\pi/4} x^{3}\sin^{6}x\,dx =\) ?
(a) \(0\)
(b) \(\pi/\sqrt{2}\)
(c) \(\pi^{3}/12\)
(d) \(2\sqrt{2}\,\pi\)
Answer: (b)
Q.163. Evaluate \(\displaystyle\int_{-\pi/2}^{\pi/2} (x^{3}+x\cos x+2\tan^{5}x+3)\,dx =\) ?
(a) \(0\)
(b) \(3\pi\)
(c) \(\pi/6\)
(d) \((\pi^{2}-2)/3\)
Answer: (b)
Q.164. Evaluate \(\displaystyle\int_{-5}^{5} [3x^{2}-x^{10}\sin x + x\sqrt{1+x^{2}}]\,dx =\) ?
(a) \(0\)
(b) \(10\)
(c) \(250\)
(d) \(375\)
Answer: (c)
Q.165. Evaluate \(\displaystyle\int_{-\pi}^{\pi} (1-x^{2})\sin x\cos^{2}x\,dx =\) ?
(a) \(0\)
(b) \(2\pi-\pi^{3}\)
(c) \(\pi/2-2\pi^{3}\)
(d) \(\pi-\pi^{3}/3\)
Answer: (b)
Q.166. Evaluate \(\displaystyle\int_{-1}^{1} \dfrac{x\sin^{-1}x}{\sqrt{1-x^{2}}}\,dx =\) ?
(a) \(0\)
(b) \(2\)
(c) \(\pi\)
(d) \(\pi/2\)
Answer: (b)
Q.167. Evaluate \(\displaystyle\int_{-1}^{1} \dfrac{x^{2}\sin^{-1}x}{\sqrt{1-x^{2}}}\,dx =\) ?
(a) \(0\)
(b) \(2\)
(c) \((\pi-2\pi^{2})/3\)
(d) \((\pi^{3}-3)/2\)
Answer: (a)
Q.168. Evaluate \(\displaystyle\int_{-\pi/2}^{\pi/2} \sin^{5}x\,dx =\) ?
(a) \(0\)
(b) \(\pi/32\)
(c) \(\pi/10\)
(d) \(2\pi/5\)
Answer: (d)
Q.169. Evaluate \(\displaystyle\int_{-a}^{a} x\sqrt{a^{2}-x^{2}}\,dx =\) ?
(a) \(0\)
(b) \(2a\)
(c) \(a^{2}/2\)
(d) \(\sqrt{2}a^{2}\)
Answer: (c)
Q.170. Evaluate \(\displaystyle\int_{-\pi}^{\pi} \dfrac{x\cos x}{1+\sin^{2}x}\,dx =\) ?
(a) \(2\pi\)
(b) \(\pi\)
(c) \(\pi/2\)
(d) \(0\)
Answer: (d)
Q.171. If \(A=\displaystyle\int_{-1}^{1}\tan x\,dx\) and \(B=\displaystyle\int_{y/2}^{y/2}\cot x\,dx\), then \(A+B=\) :
(a) \(0\)
(b) \(1\)
(c) \(\pi/4\)
(d) \(y+2\)
Answer: (b)
Q.172. Evaluate \(\displaystyle\int_{0}^{\pi/2}\dfrac{1-\cos x}{1+\cos x}\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/4\)
(d) \(\pi/2\)
Answer: (d)
Q.173. Evaluate \(\displaystyle\int_{0}^{\pi/2}(\cos x-\sin x)^{2}\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(2\)
Answer: (c)
Q.174. Evaluate \(\displaystyle\int_{0}^{\pi/2}\sqrt{1+\cos x}\,dx\).
(a) \(\sqrt{2}\)
(b) \(2\sqrt{2}\)
(c) \(\pi/\sqrt{2}\)
(d) \(\pi\)
Answer: (b)
Q.175. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/4\)
(d) \(\dfrac{\pi\ln 2}{2}\)
Answer: (d)
Q.176. Evaluate \(\displaystyle\int_{0}^{\pi/2} (\ln\sin x)(\ln\cos x)\,dx\).
(a) \(0\)
(b) \(-\dfrac{\pi}{8}(\ln 2)^{2}\)
(c) \(-\dfrac{\pi}{4}(\ln 2)^{2}\)
(d) \(\dfrac{\pi}{8}(\ln 2)^{2}\)
Answer: (c)
Q.177. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\tan x)\,dx\).
(a) \(\dfrac{\pi\ln 2}{2}\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (a)
Q.178. \(\displaystyle\int_{0}^{\pi/2}\sin^{m}x\cos^{n}x\,dx\) (m,n positive integers) result expressed by :
(a) depends on parity of \(m+n\)
(b) Beta function formula
(c) Gamma function form
(d) none
Answer: (b)
Q.179. Evaluate \(\displaystyle\int_{0}^{\pi/2}\cos^{2}x\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/4\)
(d) \(\pi/2\)
Answer: (d)
Q.180. Evaluate \(\displaystyle\int_{0}^{\pi/2}\sqrt{\sin x}\,dx\).
(a) \(0\)
(b) \(\sqrt{2}\)
(c) \(\sqrt{2}\,\dfrac{\Gamma(3/4)^{2}}{\Gamma(3/2)}\)
(d) \(\pi\)
Answer: (c)
Q.181. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin x)\,dx\).
(a) \(0\)
(b) \(-\dfrac{\pi}{2}\ln 2\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (b)
Q.182. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cos x)\,dx\).
(a) \(0\)
(b) \(-\dfrac{\pi}{2}\ln 2\)
(c) \(\pi/2\)
(d) \(\ln 2\)
Answer: (b)
Q.183. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin^{2}x)\,dx\).
(a) \(0\)
(b) \(-\pi\ln 2\)
(c) \(-\pi\ln 2\)
(d) \(1\)
Answer: (c)
Q.184. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin x\cos x)\,dx\).
(a) \(0\)
(b) \(\pi/2\)
(c) \(\pi\ln 2\)
(d) \(-\pi\ln 2\)
Answer: (d)
Q.185. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos x)\,dx\).
(a) \(0\)
(b) \(\pi\ln 2\)
(c) \(-\pi\ln 2\)
(d) \(1\)
Answer: (a)
Q.186. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\dfrac{\pi\ln 2}{2}\)
(d) \(\pi/2\)
Answer: (c)
Q.187. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\tan x)\,dx\).
(a) \(0\)
(b) \(\pi/4\)
(c) \(\dfrac{\pi\ln 2}{2}\)
(d) \(1\)
Answer: (c)
Q.188. Evaluate \(\displaystyle\int_{0}^{\pi/2}\sin x\ln(\sin x)\,dx\).
(a) \(0\)
(b) \(-1\)
(c) \(1-\ln 2\)
(d) \(\ln 2\)
Answer: (c)
Q.189. Evaluate \(\displaystyle\int_{0}^{\pi/2}\cos x\ln(\cos x)\,dx\).
(a) \(0\)
(b) \(-1\)
(c) \(1-\ln 2\)
(d) \(\ln 2\)
Answer: (c)
Q.190. Evaluate \(\displaystyle\int_{0}^{\pi/2}(\sin x-\cos x)^{2}\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/4\)
(d) \(\pi/2\)
Answer: (d)
Q.191. Evaluate \(\displaystyle\int_{0}^{\pi/2}\dfrac{\cos^{2}x}{1+\sin x}\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(2\)
Answer: (b)
Q.192. Evaluate \(\displaystyle\int_{0}^{\pi/2}\dfrac{\sin^{2}x}{1+\cos x}\,dx\).
(a) \(1\)
(b) \(2\)
(c) \(\pi/2\)
(d) \(0\)
Answer: (a)
Q.193. Evaluate \(\displaystyle\int_{0}^{\pi/2}\tan x\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\ln 2\)
(d) \(\infty\)
Answer: (d)
Q.194. Evaluate \(\displaystyle\int_{0}^{\pi/2}\cot x\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\ln 2\)
(d) \(\infty\)
Answer: (d)
Q.195. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin 2x)\,dx\).
(a) \(0\)
(b) \(\pi\)
(c) \(\dfrac{\pi\ln 2}{2}\)
(d) \(1\)
Answer: (c)
Q.196. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos 2x)\,dx\).
(a) \(0\)
(b) \(\pi\)
(c) \(-\dfrac{\pi\ln 2}{2}\)
(d) \(1\)
Answer: (c)
Q.197. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos^{2}x)\,dx\).
(a) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
(b) \(\pi\ln 2\)
(c) \(\pi/2\)
(d) \(0\)
Answer: (a)
Q.198. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin^{2}x)\,dx\).
(a) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
(b) \(\pi\ln 2\)
(c) \(\pi/2\)
(d) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
Answer: (d)
Q.199. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\tan^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi\)
(c) \(\pi\ln 2\)
(d) \(\pi\ln 2\)
Answer: (d)
Q.200. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cot^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (c)
Q.201. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sec x)\,dx\).
(a) \(0\)
(b) \(\dfrac{\pi}{2}\ln 2\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (b)
Q.202. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\csc x)\,dx\).
(a) \(0\)
(b) \(\pi\)
(c) \(\dfrac{\pi}{2}\ln 2\)
(d) \(\ln 2\)
Answer: (c)
Q.203. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\tan x)\,dx\).
(a) \(0\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(\ln 2\)
Answer: (b)
Q.204. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cot x)\,dx\).
(a) \(0\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.205. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\sin x)\,dx\).
(a) \(0\)
(b) \(-\pi^{2}/12\)
(c) \(-\dfrac{\pi^{2}}{8}\ln 2\)
(d) \(\pi/2\)
Answer: (c)
Q.206. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\cos x)\,dx\).
(a) \(-\pi^{2}/12\)
(b) \(-\pi^{2}/8\)
(c) \(\pi/2\)
(d) \(0\)
Answer: (a)
Q.207. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\sec x)\,dx\).
(a) \(0\)
(b) \(\pi^{2}/12\)
(c) \(\pi^{2}/8\)
(d) \(1\)
Answer: (b)
Q.208. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\csc x)\,dx\).
(a) \(0\)
(b) \(\pi^{2}/12\)
(c) \(\pi^{2}/8\)
(d) \(1\)
Answer: (b)
Q.209. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\tan x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(0\)
(d) \(\ln 2\)
Answer: (c)
Q.210. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\cot x)\,dx\).
(a) \(0\)
(b) \(0\)
(c) \(1\)
(d) \(\ln 2\)
Answer: (b)
Q.211. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin x\cos x)\,dx\).
(a) \(-\pi\ln 2\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (a)
Q.212. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi/2\)
(c) \(-\pi\ln 2\)
(d) \(-\pi\ln 2\)
Answer: (d)
Q.213. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin x\cos x)\,dx\).
(a) \(-\pi\ln 2\)
(b) \(\pi/2\)
(c) \(0\)
(d) \(1\)
Answer: (a)
Q.214. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cos^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi\ln 2\)
(c) \(-\pi\ln 2\)
(d) \(1\)
Answer: (c)
Q.215. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin 2x)\,dx\).
(a) \(0\)
(b) \(\dfrac{\pi\ln 2}{2}\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.216. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos 2x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(-\dfrac{\pi\ln 2}{2}\)
Answer: (d)
Q.217. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (b)
Q.218. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin^{2}x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
Answer: (d)
Q.219. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\tan^{2}x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi\ln 2\)
(d) \(\pi/2\)
Answer: (c)
Q.220. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cot^{2}x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(\pi\ln 2\)
Answer: (d)
Q.221. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sec x)\,dx\).
(a) \(\dfrac{\pi}{2}\ln 2\)
(b) \(0\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (a)
Q.222. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\csc x)\,dx\).
(a) \(\dfrac{\pi}{2}\ln 2\)
(b) \(0\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (a)
Q.223. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\tan x)\,dx\).
(a) \(\pi/2\)
(b) \(1\)
(c) \(0\)
(d) \(\ln 2\)
Answer: (c)
Q.224. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cot x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(\ln 2\)
Answer: (a)
Q.225. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\sin x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(-\dfrac{\pi^{2}}{8}\ln 2\)
Answer: (d)
Q.226. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\cos x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(-\dfrac{\pi^{2}}{12}\)
(d) \(\pi/2\)
Answer: (c)
Q.227. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\sec x)\,dx\).
(a) \(\pi^{2}/12\)
(b) \(0\)
(c) \(\pi^{2}/8\)
(d) \(1\)
Answer: (a)
Q.228. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\csc x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(\pi^{2}/12\)
Answer: (d)
Q.229. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\tan x)\,dx\).
(a) \(0\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.230. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\cot x)\,dx\).
(a) \(0\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.231. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin x\cos x)\,dx\).
(a) \(-\pi\ln 2\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (a)
Q.232. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi/2\)
(c) \(-\pi\ln 2\)
(d) \(1\)
Answer: (c)
Q.233. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cos^{2}x)\,dx\).
(a) \(0\)
(b) \(-\pi\ln 2\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.234. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin 2x)\,dx\).
(a) \(\dfrac{\pi\ln 2}{2}\)
(b) \(0\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (a)
Q.235. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos 2x)\,dx\).
(a) \(0\)
(b) \(-\dfrac{\pi\ln 2}{2}\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.236. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cos^{2}x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
(d) \(\pi/2\)
Answer: (c)
Q.237. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\sin^{2}x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi\ln\!\dfrac{1+\sqrt{2}}{2}\)
(d) \(\pi/2\)
Answer: (c)
Q.238. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\tan^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi\ln 2\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.239. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(1+\cot^{2}x)\,dx\).
(a) \(0\)
(b) \(\pi\ln 2\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.240. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sec x)\,dx\).
(a) \(\dfrac{\pi}{2}\ln 2\)
(b) \(0\)
(c) \(\pi\ln 2\)
(d) \(1\)
Answer: (a)
Q.241. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\csc x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\dfrac{\pi}{2}\ln 2\)
(d) \(\pi\ln 2\)
Answer: (c)
Q.242. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\tan x)\,dx\).
(a) \(\pi/2\)
(b) \(\ln 2\)
(c) \(1\)
(d) \(0\)
Answer: (d)
Q.243. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cot x)\,dx\).
(a) \(\pi/2\)
(b) \(0\)
(c) \(\ln 2\)
(d) \(1\)
Answer: (b)
Q.244. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\sin x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(-\dfrac{\pi^{2}}{8}\ln 2\)
Answer: (d)
Q.245. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\cos x)\,dx\).
(a) \(0\)
(b) \(-\dfrac{\pi^{2}}{12}\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.246. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\sec x)\,dx\).
(a) \(\pi^{2}/12\)
(b) \(0\)
(c) \(\pi^{2}/8\)
(d) \(1\)
Answer: (a)
Q.247. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\csc x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(\pi^{2}/12\)
Answer: (d)
Q.248. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\tan x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(0\)
(d) \(\ln 2\)
Answer: (c)
Q.249. Evaluate \(\displaystyle\int_{0}^{\pi/2} x\ln(\cot x)\,dx\).
(a) \(0\)
(b) \(1\)
(c) \(\pi/2\)
(d) \(\ln 2\)
Answer: (a)
Q.250. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin x\cos x)\,dx\).
(a) \(\pi/2\)
(b) \(0\)
(c) \(\ln 2\)
(d) \(-\pi\ln 2\)
Answer: (d)
Q.251. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\sin^{2}x)\,dx\).
(a) \(0\)
(b) \(-\pi\ln 2\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)
Q.252. Evaluate \(\displaystyle\int_{0}^{\pi/2}\ln(\cos^{2}x)\,dx\).
(a) \(0\)
(b) \(-\pi\ln 2\)
(c) \(\pi/2\)
(d) \(1\)
Answer: (b)

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