Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The value of the integral  <img alt="\mathrm{\int_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}}" src="https://entrancecorner.oncodecogs.com/gi

The value of the integral  \mathrm{\int_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}} is equal to

Option: 1

2\pi


Option: 2

0


Option: 3

\pi


Option: 4

\frac{\pi}{2}


Answers (1)

best_answer

\mathrm {I=\int_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}}..................(i)

also

\mathrm {I=\int_{-\pi / 2}^{\pi / 2} \frac{e^{x} d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)} }..................(ii)

on adding qeuation (i) and (ii)

\mathrm{2 I=\int_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(\sin ^{6} x+\cos ^{6} x\right)}}

\mathrm{2 I=2 \int_{0}^{\pi / 2} \frac{d x}{\left(\sin ^{4} x-\sin ^{2} x \cos ^{2} x+\cos ^{4} x\right)}}

\mathrm{I=\int_{0}^{\pi / 2} \frac{\left(1+\tan ^{2} x\right) \sec ^{2} x d x}{\left(\tan ^{4} x-\tan ^{2} x+1\right)} }

Let \mathrm{\tan x=t}

\begin{aligned} & \mathrm{I=\int_{0}^{\infty} \frac{1+t^{2}}{\left(t+-t^{2}+1\right)} d t }\\ &\mathrm{I=\int_{0}^{\infty} \frac{\left(1+\frac{1}{t^{2}}\right) d t}{\left(t-\frac{1}{t}\right)^{2}+1} }\end{aligned}

let   \mathrm{t-\frac{1}{t}=z}

\mathrm{ I=\int_{-\infty}^{\infty} \frac{d z}{z^{2}+1} =\left[\tan ^{-1} z\right]_{-\infty}^{\infty}} \\

    \mathrm{=\pi} \\

 

Posted by

jitender.kumar

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

Bihar Board extends BSEB Super 50 free coaching application last date to March 26
anam-khan

JEE Main 2025: Most important chapters for physics, chemistry, mathematics to score good marks
anam-khan

JEE Main 2025 session 2 dates clash with ISC, PSEB 12th exams; no impact on major science subjects

Latest Question