Question

What is the antiderivative of `ln(x^2 + 1)`?

Answer 1

The antiderivative is the integral

`int ln(x^2+1)dx`

Using integration by parts we get

#int ln(x^2+1)dx=int x' ln (x^2+1)dx=x*ln(x^2+1)-int x[2x/(x^2+1)]dx=x*ln(x^2+1)-2 int x^2/(1+x^2)dx= x*ln(x^2+1)-2int [(x^2+1-1)/(x^2+1)]dx= x*ln(x^2+1)-2*x+2*arctanx+c #

Finally

#int ln(x^2+1)dx= x*ln(x^2+1)-2*x+2*arctanx+c #