Question

x3+x+1x2-112.

Answer 1

The integral is given below as,

I= ∫ x 3 +x+1 x 2 −1 dx

As the given integral is not a proper fraction. Divide ( x 3 +x+1 )by ( x 2 −1 ).

x 3 +x+1 x 2 −1 =x+ 2x+1 ( x+1 )( x−1 )

Use rule of partial fraction.

2x+1 ( x+1 )( x−1 ) = A x+1 + B x−1 2x+1=A( x−1 )+B( x+1 )

Substitute x=1then,

B= 3 2

Now, substitute x=−1 then,

A= 1 2

On integrating, we get

I= ∫ x 3 +x+1 x 2 −1 dx = ∫ xdx+ ∫ 2x+1 ( x+1 )( x−1 ) dx = x 2 2 + 1 2 ∫ dx ( x+1 ) + 3 2 ∫ dx x−1 = x 2 2 + 1 2 log| x+1 |+ 3 2 log| x−1 |+C