u-Substitution: How to Integrate Composite Functions
Learn u-substitution with two worked examples and the most common mistakes.
The idea
- ∫ f(g(x)) g′(x) dx = ∫ f(u) du, with u = g(x)
Pick u as the inside piece whose derivative is also in the integral. Then swap everything to u.
Example 1
Evaluate ∫ 2x (x² + 1)³ dx
- u = x² + 1, du = 2x dx
- ∫ u³ du = u⁴ / 4 + C
- Substitute back: u = x² + 1
(x² + 1)⁴ / 4 + C
Example 2
Evaluate ∫ cos(3x) dx
- u = 3x, du = 3 dx, so dx = du / 3
- (1/3) ∫ cos u du = (1/3) sin u + C
(1/3) sin(3x) + C
Common mistake
∫ cos(3x) dx = sin(3x) + C
Do not forget the constant factor from du. Here it is 1/3.
Never leave x and u mixed in one integral. Convert everything, then back-substitute.
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