Evaluate the Integral (using Integration By Parts)

Question

Evaluate the Integral (using Integration By Parts). Show a step by step solution. If possible could you make a video to go through the problem? the final answer is \frac{2\sqrt{3}\ln \left(2\right)+2\pi -\sqrt{3}\pi }{4\sqrt{3}}

Answer 1

Let u = Arctan(1/x) and dv = dx.

Then du = [1 / (1 + (1/x²))] (-1/x²)dx and v = x.

Simplifying, we get du = -1 / (1 + x²) dx and v = x.

So, ∫Arctan(1/x)dx = xArctan(1/x) + ∫[x / (1 + x²) ]dx = xArctan(1/x) + (1/2) ln(1+x²) + C

Evaluating from 1 to √3, we have: √3Arctan(1/√3) + (1/2)ln4 - Arctan(1) - (1/2)ln2

= √3(π/6) - π/4 + (1/2) ln2