Question

[Hint: multiply numerator and denominator by x-1 and put x-t ]

Answer 1

The integral is given as follows,

I= ∫ dx x( x n +1 )

Multiply and divide the given integral by x n−1 .

I= ∫ x n−1 dx x n−1 ×x( x n +1 ) = ∫ x n−1 dx x n ( x n +1 )

Substitute x n =t

Differentiate with respect to t.

n x n−1 dx=dt x n−1 dx= dt n

Substitute the values.

I= ∫ dx x( x n +1 ) I= ∫ x n−1 dx x n ( x n +1 ) = 1 n ∫ dt t( t+1 )

Use partial fraction to reduce function.

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

Substitute t=−1.

B=−1

Substitute t=0.

A=1

Now substitute the values.

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

Substitute x n for t.

I= 1 n [ log| x n |−log| x n +1 | ]+C I= 1 n log| x n x n +1 |+C