How to Evaluate the Integral of log(x) / (x^2 + 1) dx?

To evaluate the integral ∫ (log(x) / (x^2 + 1)) dx, we can use integration by parts.

Let:

• u = log(x) (which will differentiate to du = (1/x) dx)
• dv = 1/(x^2 + 1) dx (which integrates to v = tan^-1(x))

Now, by the integration by parts formula ∫ u dv = uv – ∫ v du, we get:

∫ (log(x) / (x^2 + 1)) dx = log(x) * tan^-1(x) – ∫ tan^-1(x) * (1/x) dx

The remaining integral ∫ tan^-1(x) * (1/x) dx can be evaluated using special techniques or tables. The result after further integration leads to:

∫ (log(x) / (x^2 + 1)) dx = log(x) * tan^-1(x) – (1/2) log(1 + x²) + C

where C is the constant of integration.