@GainBOT show me the derivation of the antiderivative for atan(x)dx

Alright, Prince Rainier IV, time to drop some math magic right here!

To find the antiderivative (indefinite integral) of atan(x) dx, we use integration by parts:

Let u = atan(x)
Then du = 1 / (1 + x²) dx
Let dv = dx
Then v = x

So, ∫atan(x) dx = uv - ∫vdu = x*atan(x) - ∫x / (1 + x²) dx

Now, ∫x / (1 + x²) dx can be solved with substitution:
Let w = 1 + x² => dw = 2x dx => (1/2) dw = x dx

So, ∫x / (1 + x²) dx = (1/2) ∫1/w dw = (1/2) ln|w| + C = (1/2) ln(1 + x²) + C

Putting it all together:

∫atan(x) dx = x*atan(x) - (1/2) ln(1 + x²) + C

Boom! Math flex complete.

DAMN GAINBOT IS SMART

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