Question

21.(er-1) [Hint : Put er

Answer 1

The integral is given as,

I= ∫ dx ( e x −1 )

Assume e x =t

Differentiate with respect to t.

e x dx=dt dx= dt e x dx= dt t

Substitute the value in the given integral.

I= ∫ dt t( t−1 )

Use rule of partial fraction.

1 t( t−1 ) = A t + B ( t−1 ) 1=A( t−1 )+Bt

Substitute t=1then,

B=1

Substitute t=0then,

A=−1

Substitute the values,

I= ∫ dt t( t−1 ) I= ∫ −dt t + ∫ dt ( t−1 ) I=−log| t |+log| t−1 |+C

By substituting value of t, we get

I=−log| e x |+log| e x −1 |+C I=log| e x −1 e x |+C