Question

How do you integrate arctan(x dx? i think that if you simplify the integral you get:?

Answer 1

Integration by parts will help here. Letting
`u=\arctan x` and
`\mathrm dv=\mathrm dx`, you end up with
`\mathrm du=(\mathrm dx)/(1+x^2)` and
`v=x`. Now


`\displaystyle\int\arctan x\,\mathrm dx=uv-\int v\,\mathrm du`

`\displaystyle\int\arctan x\,\mathrm dx=x\arctan x-\int\frac x{1+x^2}\,\mathrm dx`

For the remaining integral, setting
`y=1+x^2` gives
`\frac{\mathrm dy}2=x\,\mathrm dx`, so


`\displaystyle\int\frac x{1+x^2}\,\mathrm dx=\frac12\int\frac{\mathrm dy}y=\frac12\ln|y|+C=\frac12\ln(1+x^2)+C`

Putting everything together, you end up with


`\displaystyle\int\arctan x\,\mathrm dx=x\arctan x-\frac12\ln(1+x^2)+C`