Integration Tips: Spotting Which Method to Use

Differentiating is mechanical. Integrating is detective work. The good news is that most integration problems fall into a small number of patterns.

Pattern 1: standard polynomials

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C

If you see a polynomial, this is your tool. Don't forget the +C.

Pattern 2: substitution (the inner-function trick)

If you see a function inside another function, with the derivative of the inner one nearby, use substitution.

Example: ∫ 2x · cos(x²) dx. The 2x is the derivative of x². Let u = x². Then du = 2x dx. The integral becomes ∫ cos(u) du = sin(u) + C = sin(x²) + C.

Pattern 3: integration by parts

If you see a product of two unrelated functions, try integration by parts.

∫ u dv = uv - ∫ v du

The trick is choosing u and dv. Pick u to be something that gets simpler when differentiated (logs, polynomials). Pick dv to be something easy to integrate (exponentials, trig).

LIATE rule for picking u: Logarithms, Inverse trig, Algebraic, Trig, Exponential — in that order.

Pattern 4: partial fractions

If you see a rational function (polynomial over polynomial) with the denominator factorable, split it into simpler fractions, integrate each separately.

Common pitfalls

• Forgetting +C
• Wrong sign on negative substitution
• Not checking by differentiating the answer (free check, do it every time)

When you're stuck

Try in this order:

1. Recognise standard form
2. Try substitution
3. Try integration by parts
4. Try partial fractions
5. Look it up — some integrals don't have elementary forms

Try a worked integral →