Make a substitution to express the integrand as a rational function and then evaluate the integral

Question

Make a substitution to express the integrand as a rational function and then evaluate the integral

∫ 5 / x(sqrt(x+1)𝑑𝑥

Answer 1

The substitution can be u = sqrt(x = 1)

This leads to an integrand 5/(u^2 -1 ) = 5/ [(u-1)(u+1) ]

This can be worked out using decomposition into partial fractions.

Answer 2

let u = ( x+1 ) ^1/2

= 5 integral ( 2/ u^2 -1) du

= 5*2 integral ( 1/ u^2 -1) du

common integral

integral ( 1/ u^2 -1) du = ln|u+1|/2 - ln|u-1|/2|

= -5[ ln|( x+1 )^1/2 +1| - ln|( x+1 )^1/2 -1|] +C