Question

What is the indefinite integral of `1/(1+sqrt(x+1))`?

Answer 1

`int1/(1+sqrt(x+1))dx=2sqrt(x+1)-2ln|1+sqrt(x+1)|+C`

Explanation:

Let

`I=int1/(1+sqrt(x+1))dx`

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

`I=2intu/(1+u)du`

Rearrange:

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

Integrate directly:

`I=2(u-ln|1+u|)+C`

Reverse the substitution:

`I=2sqrt(x+1)-2ln|1+sqrt(x+1)|+C`