logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Model Papers
0 votes

Evaluate :$ \int_0^{\pi}\large \frac{e^{\large \cos x}}{e^{\large \cos x}+e^{\large-\cos x}}$$dx$

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • (i) $\int \limits_a^b f(x)dx=F(b)-F(a)$
  • (ii) $ \int \limits_a^b f(x)dx=\int \limits_a^b f(a-x) dx$
  • (iii) $ \cos (\pi-x)=\cos x$
Step 1:
Given $\int \limits _0 ^ \pi \Large\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}$$dx$------(1)
By applying the property $ \int \limits_a^b f(x)dx=\int \limits_a^b f(a-x) dx$
$I=\int \limits _0 ^ \pi \Large\frac{e^{\cos (\pi-x)}}{e^{\cos (\pi-x)}+e^{-\cos (\pi-x)}}$$dx$
But $\cos (\pi-x)=-\cos x $
Therefore $I=\int \limits _0 ^ \pi \Large\frac{e^{-\cos x}}{e^{-\cos x}+e^{-(-\cos x)}}$$dx=\int \limits _0 ^ \pi \Large\frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}}$$dx$------(2)
Step 2:
Adding equ(1) and equ (2)
$2I=\int \limits _0 ^ \pi \Large\frac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}+\frac{e^{-\cos x}}{e^{-\cos x}+e^{\cos x}}$$dx$
$\quad=\int \limits _0 ^ \pi \Large\frac{e^{\cos x}+e ^{-\cos x}}{e^{\cos x}+e^{-\cos x}}$$dx$
$\quad=\int \limits _0 ^ \pi dx$
Step 3:
on integrating we get
$2I=\big[x\big]_0^\pi$
on applying limits we get,
$2I=\pi-0=\pi$
Therefore $ I=\frac{\pi}{2}$
answered Sep 24, 2013 by sreemathi.v
 
Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...