Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

The value of    \int_{-\pi }^{\pi }\frac{cos^{2}x}{1+a^{x}}\, dx,a> 0,\; \; is 

  • Option 1)

    \pi /2\;

  • Option 2)

    \; \; a\pi \;

  • Option 3)

    \; \; 2\pi \;

  • Option 4)

    \; \; \pi /a

 

Answers (1)

best_answer

As we learnt in 

Properties of Definite Integration -

\int_{0}^{2a}f(x)dx= \int_{0}^{a}\left [ f(x)+f(-x) \right ]dx

= \left\{\begin{matrix} 2\int_{0}^{a}f(x)dx & if f(2a-x)=f(x)\\ 0 &if f(2a-x)=-f(x) \end{matrix}\right.

-

 

 I= \int_{-\pi}^{\pi}\frac{\cos^2x}{1+a^x}

I= \int_{0}^{\pi}\left ( \frac{\cos^2x}{1+a^x}+\cos^2x\frac{a^x}{1+a^x} \right )dx

I= \int_{0}^{\pi} \left(cos^{2}x \right )dx

I= \int_{0}^{\pi}\left ( \frac{1+\cos2x}{2} \right )dx

 I= \frac{\pi}{2}+\frac{1}{2}\int_{0}^{\pi}\cos2x \:dx

= \frac{\pi}{2}+\frac{1}{2}\left [\frac{\sin2x }{2} \right ]_{0}^{\pi}

= \frac{\pi}{2}+\frac{1}{2}\times0=\frac{\pi}{2}


Option 1)

\pi /2\;

This is correct option

Option 2)

\; \; a\pi \;

This is incorrect option

Option 3)

\; \; 2\pi \;

This is incorrect option

Option 4)

\; \; \pi /a

This is incorrect option

Posted by

divya.saini

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

JEE Main 2024 session 2 final answer key soon at jeemain.nta.ac.in; result on April 25
anam-khan

JEE Main 2024 Session 2: Last day to challenge answer key today
anam-khan

BTech admission 2024 without JEE; top government engineering colleges, upcoming entrance exams

Latest Question