Question

integrate log(1+x^2)

Answer 1

X log X (1+x.sq)-2x+2tan inverse X+C

Answer 2

We have to find the indefinite integral $\int \ln(1+x^2) \, dx$.

Use integration by parts,

$\int \ln(1+x^2) \, dx = \int 1.\ln(1+x^2) \, dx \\ \int \ln(1+x^2) \, dx =x\ln(1+x^2)- \int \frac{2x^2}{1+x^2} \, dx \\ \int \ln(1+x^2) \, dx =x\ln(1+x^2)- 2\int (1-\frac{1}{1+x^2}) \, dx \\ \int \ln(1+x^2) \, dx =x\ln(1+x^2)- 2x+2\tan^{-1}x+C$