Short on time? This page is the recap. For explanations, the area diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) f(x) dx is the definite integral from x = a to x = b.
Recap
- Substitution is the chain rule backwards. It works when the integrand is (outer function of an inner function) × (derivative of the inner function), up to a constant.
- Steps: choose u, find du, rewrite everything in u, integrate, substitute back, add + C, check by differentiating.
- Constants can be adjusted (du = 2x dx gives x dx = du/2). Variables cannot.
- If an x is left over, solve u = g(x) for x and replace it.
- In a definite integral, change the limits to u = g(a) and u = g(b) and finish in u. Or go back to x first. Never mix the two.
Key relationships
| Idea | Rule |
|---|---|
| Substitution rule | ∫ f(g(x)) g′(x) dx = F(g(x)) + C, where F′ = f |
| Linear inner function | ∫ f(ax + b) dx = (1/a) F(ax + b) + C |
| Changing limits | ∫ (a to b) f(g(x)) g′(x) dx = ∫ (g(a) to g(b)) f(u) du |
| Derivative over function | ∫ g′(x)/g(x) dx = ln |
| A standard result | ∫ tan x dx = −ln |
Typical choices of u: the base of a power, the inside of a root, the exponent of e, the input of sin or cos, a denominator whose derivative is in the numerator.
Assumptions behind the method
- u = g(x) must be differentiable, with g′ continuous, on the interval you integrate over.
- The outer function f must be continuous on the u-values that g takes there.
- The integral must become a function of u alone, with nothing left in x.
Mistakes to avoid
- Missing the constant factor (for example, forgetting the 1/2 when du = 2x dx).
- Dividing by a variable to make du appear.
- Keeping x-limits with an antiderivative in u.
- Swapping limits that come out in the “wrong” order, without changing the sign.
- Forgetting + C on an indefinite integral.
- Leaving x and u mixed in the same integral.
Quick self-check
- Find ∫ 3x² e^(x³) dx. (e^(x³) + C, with u = x³)
- Find ∫ sin(4x) dx. (−(1/4) cos(4x) + C)
- Evaluate ∫ (0 to 2) x(x² + 1)² dx. (62/3: with u = x² + 1 the integral is (1/2) ∫ (1 to 5) u² du = (1/2)(125 − 1)/3)
Next: practice questions.