How do you intagrate this function?

(1+sqrt((x+1) ) / 1-sqrt((x+1))

1 Answer
Jun 9, 2018

Use the substitution 1-sqrt(x+1)=u.

Explanation:

Let

I=int(1+sqrt(x+1))/(1-sqrt(x+1))dx

Apply the substitution 1-sqrt(x+1)=u:

I=int(2-u)/u(-2(1-u)du)

Simplify:

I=int(6-2u-4/u)du

Integrate directly:

I=6u-u^2-4ln|u|+C

Reverse the substitution:

I=(5+sqrt(x+1))(1-sqrt(x+1))-4ln|1-sqrt(x+1)|+C

Simplify and rescale C:

I=-x-4sqrt(x+1)-4ln|1-sqrt(x+1)|+C