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

1 Answer
May 23, 2018

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