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Q21 Integrate the rational functions \frac{1}{( e ^x-1)} [Hint : Put e ^x= t]

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Given function \frac{1}{( e ^x-1)}

So, applying the hint: Putting e^x = t

Then e^x dx= dt

\int \frac{1}{( e ^x-1)}dx = \int\frac{1}{t-1}\times\frac{dt}{t} = \int \frac{1}{t(t-1)}dt

 

Partial fraction of above equation,

\frac{1}{t(t-1)} = \frac{A}{t}+\frac{B}{(t-1)}

1= A(t-1)+Bt                                            ..............(1)                            

Now, substituting t = 0\ and\ t = 1 in equation (1), we get

A = -1\ and\ B=1

\Rightarrow \frac{1}{t(t+1)} = -\frac{1}{t}+\frac{1}{t-1}

\implies \int \frac{1}{t(t-1)}dt = \log \left | \frac{t-1}{t} \right |+C

Now, back substituting the value of t,

= \log \left | \frac{e^x-1}{e^x} \right |+C

 

Posted by

Divya Prakash Singh

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