How do you find the integral of ln(1+x^2)?

1 Answer
Jun 3, 2018

The answer is =xln(1+x^2)-2x+2arctanx+C

Explanation:

Calculate this integral by integration by parts

intuv'=uv-intu'v

u=ln(1+x^2), =>, u'=(2x)/(1+x^2)

v'=1, =>, v=x

Therefore, the integral is

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

The second integral is

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

=2int1dx-2int(1dx)/(1+x^2)

=2x-2arctanx

Finally, the integral is

I=xln(1+x^2)-2x+2arctanx+C