Integrate by Parts: Solving \int \ln (x^2 + 1) \, dx

[GunnaSix]

Homework Statement

Find or evaluate the integral using substitution first, then using integration by parts.

$$\int \ln (x^2 + 1) \, dx$$

The Attempt at a Solution

$$Let \: u = x^2 + 1$$

$$du = 2x \, dx$$

$$dx = \pm \frac{du}{2 \sqrt{u - 1}}$$

Then

$$\int \ln (x^2 + 1) \, dx = \pm \frac{1}{2} \int \frac{\ln u}{\sqrt{u-1}}\, du$$

I don't know where to go from here. I tried to integrate by parts and it just turned into a mess. Am I approaching this the wrong way?

 

[Dick]

I'm not sure what substitution they think you ought to make. It's pretty easy if you apply parts right off regarding the original integral as u*dv where u=ln(1+x^2) and v=x.

 

[GunnaSix]

Got it. Thanks.