Quick Overview
Integration by Substitution
Simplify complex integrals by substituting u = f(x) and converting everything in terms of u
Harder Substitution
When a substitution is given, apply the same process but expect trickier algebra
Integration by Parts
For products of functions using ∫u dv/dx dx = uv - ∫v du/dx dx
Partial Fractions
Split complex fractions into simpler pieces that integrate easily
Volumes of Revolution
Find volumes when 2D regions rotate around an axis using V = π∫y²dx
Decision Making
Use a systematic approach to choose which integration technique fits each problem
1. Integration by Substitution (Reverse Chain Rule)
When a function is too awkward to integrate directly, and you can't spot the reverse chain rule immediately, substitution transforms the integral into something manageable.
What is Integration by Substitution?
Substitution is useful when:
- The reverse chain rule is hard to spot
- You have a composite function (a function inside another function)
- The substitution is "obvious" from context—you're expected to find it
The key insight: Look to substitute the "second" (inner) function rather than the "main" (outer) function.
Example: In ∫18x(x² + 2)⁵ dx, the main function is (...)⁵ and the second function is x² + 2
So u = x² + 2 would be a suitable substitution.
The Four-Step Method
STEP 1: Determine the substitution
Ask yourself: What is the 'main' function? What is the 'second' function? Choose the second one as u.
STEP 2: Differentiate the substitution and rearrange
Find du/dx, then treat it like a fraction. Rearrange to get dx in terms of du.
Example: If u = 5x³, then du/dx = 15x², so dx = du/(15x²)
STEP 3: Replace all parts of the integral
Replace all x terms with u terms (including dx).
If it's a definite integral, change the limits from x values to u values.
STEP 4: Integrate and substitute back
The result should be easy to integrate now. After integrating, substitute x back in.
Don't forget the constant c for indefinite integrals!
If u = f(x), then ∫g(f(x))·f'(x)dx = ∫g(u)du
Worked Example
Find: ∫₀¹ 15x² cos(5x³) dx
STEP 1: DETERMINE THE SUBSTITUTION
u = 5x³ (the inner function)
STEP 2: DIFFERENTIATE AND REARRANGE
du/dx = 15x²
du = 15x² dx
STEP 3: REPLACE ALL PARTS
When x = 0: u = 0
When x = 1: u = 5
∫₀¹ 15x² cos(5x³) dx = ∫₀⁵ cos(u) du
STEP 4: INTEGRATE AND EVALUATE
= [sin(u)]₀⁵
= sin(5) - sin(0)
= sin(5) ≈ -0.959 (3 s.f.)
Examiner Tip:
Practice Questions