How would you integrate ln(x^2 + 1)?

1 Answer
Jul 27, 2018

int ln(x^2+1) dx = xln(x^2+1)- 2 x+2arctanx+C

Explanation:

Integrate by parts:

int ln(x^2+1) dx = int ln(x^2+1) d/dx (x) dx

int ln(x^2+1) dx = xln(x^2+1)- int x d/dx( ln(x^2+1) ) dx

int ln(x^2+1) dx = xln(x^2+1)- 2 int x^2/(x^2+1) dx

Now:

int ln(x^2+1) dx = xln(x^2+1)- 2 int (x^2+1-1)/(x^2+1) dx

and using the linearity of the integral:

int ln(x^2+1) dx = xln(x^2+1)- 2 int dx+2int 1/(x^2+1) dx

These are well known integrals, so:

int ln(x^2+1) dx = xln(x^2+1)- 2 x+2arctanx+C