Integration by substitution and by parts

When simple rules don’t work, two techniques help: substitution (the reverse of the chain rule) and integration by parts (the reverse of the product rule).

Key facts

  • Substitution: choose u, replace dx using du/dx, change limits too.
  • By parts: ∫ u dv = uv − ∫ v du.
  • Choose u to be the part that gets simpler when differentiated (LIATE: logs, inverse trig, algebra, trig, exponentials).
  • Partial fractions first for rational functions.

Worked examples

Example 1

Find ∫ x eˣ dx.

  1. u = x, dv = eˣ dx, so du = dx, v = eˣ.
  2. x eˣ − ∫ eˣ dx = x eˣ − eˣ + c

Answer: x eˣ − eˣ + c

Example 2

Find ∫ 2x(x² + 1)⁴ dx.

  1. Let u = x² + 1, du = 2x dx.
  2. ∫ u⁴ du = u⁵/5 + c
  3. = (x² + 1)⁵/5 + c

Answer: (x² + 1)⁵/5 + c