How do you evaluate the integral int sqrt(e^x+1)?
1 Answer
May 23, 2018
Use the substitution
Explanation:
Let
I=intsqrt(e^x+1)dx
Apply the substitution
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