3.5.1—Extended integration rules

Use substitution to expose an inner derivative, integration by parts to reduce a product, and partial fractions to split a rational function into standard terms.

Inspect the integrand before starting: ask what becomes simpler after differentiating one factor or replacing a repeated expression. Change limits and differentials consistently.

For ∫x ln x dx, take u=ln x and dv=x dx; for ∫2x/(1+x²)dx, use u=1+x².

A method is not justified because it produces more terms; the check is whether the new integral is genuinely simpler.