Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Integrals
0 votes

Integrate the function $\int\large\frac{e^x}{(1+e^x)(2+e^x)}$

Can you answer this question?

1 Answer

0 votes
  • $\int \large\frac{1}{(x+a)}$$dx=\log|x+a|+c$
Step 1:
Let $I=\int \large\frac{e^x}{(1+e^x)(2+e^x)}$$dx$
On differentiating with respect to $x$ we get,
$I=\int\large\frac{e^x dt}{(1+t)(2+t)e^x}$
$\;\;\;=\int\large\frac{ dt}{(1+t)(2+t)}$
Step 2:
Let $\large\frac{1}{(1+t)(2+t)}=\frac{A}{(1+t)}+\frac{B}{(2+t)}$-----(1)
$\Rightarrow 1=A(2+t)+B(1+t)$
$\Rightarrow (2A+B)+t(A+B)$
On equating the coefficients of $t$ and constant term on both sides we get
$(A+B)=0$ and $2A+B=1$
Step 3:
On solving both equations we get,
Step 4:
Substitute the value of A and B in equ(1)
$I=\int \large\frac{1}{1+t}$$dt-\int \large\frac{1}{2+t}$$dt$
Differentiating the above equation with respect to $t$ we get,
$I=\log \mid 1+t\mid -\log \mid 2+t\mid +c$
answered Sep 10, 2013 by sreemathi.v
Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App