Question

Evaluate: `int (2x)/(1 + x^2) dx`

Answer 1

We need to evaluate:

`int (2x)/(1 + x^2) dx`

Step 1: Substitution

Let u = 1 + x²

Then, `(du)/dx = 2x`

du = 2x dx

Step 2: Substitute into the integral

`int (2x)/(1 + x^2) dx = int (du)/u`

Step 3: Integrate

`int 1/u du = ln |u| + C`

Substitute back u = 1 + x²:

= ln |1 + x²| + C

Since 1 + x² > 0 for all real x, absolute value is optional.

ln (1 + x²) + C