Question

How do you evaluate the integral `int sqrt(e^x+1)`?

Answer 1

Use the substitution `sqrt(e^x+1)=u`.

Explanation:

Let

`I=intsqrt(e^x+1)dx`

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

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

Rearrange:

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

Factorize the denominator:

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

Apply partial fraction decomposition:

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

Integrate term by term:

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

Reverse the substitution:

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