what is the integral of xarctan(5x)dx?

Answer 1

Use integration by parts:

Let u = arctan(5x)
dv = xdx

Then du = 5dx/(1 + 25x²)
v = x²/2

Note:
v du = (5/2) x²/(1 + 25x²) dx
= (1/10) [25x²/(1 + 25x²)]
= (1/10)[1 + 25x² - 1]/(1 + 25x²)]
= (1/10)[1 - 1/(1 + 25x²)]