### Integrate Rational Function with Quadratic Denominator

solve ∫xx2+1dx\int \frac{x}{x^2 + 1} \, dx∫x2+1x​dx

∫xx2+1dx=12ln⁡∣x2+1∣+C\int \frac{x}{x^2 + 1} \, dx = \frac{1}{2} \ln\|x^2 + 1\| + C∫x2+1x​dx=21​ln∣x2+1∣+C

**Quick derivation:** Let u=x2+1u = x^2 + 1u=x2+1, so du=2xdxdu = 2x\,dxdu=2xdx. The integral becomes 12∫duu=12ln⁡∣u∣+C\frac{1}{2}\int \frac{du}{u} = \frac{1}{2}\ln\|u\| + C21​∫udu​=21​ln∣u∣+C.