Math · Calculus I · Concept
Integration by substitution (u-substitution)
Integration by substitution reverses the chain rule. When an integrand contains an inner function g(x) and its derivative g′(x), set u = g(x), so du = g′(x) dx; integrate in u, then substitute back. For a definite integral, change the limits to u-values instead.
The chain rule, backward
The chain rule says the derivative of F(g(x)) is F′(g(x))·g′(x). Read backward, an integrand of the form f(g(x))·g′(x) has antiderivative F(g(x)) + C, where F′ = f. Substitution is the bookkeeping that makes this pattern visible.
Choosing u
Look for an inner function whose derivative also appears as a factor, perhaps up to a constant. The inside of a power, a root, an exponent or a trig function is usually the right choice.
| Integrand | u | du |
|---|---|---|
| 2x(x² + 1)⁵ | x² + 1 | 2x dx |
| x e^(x²) | x² | 2x dx |
| x²√(x³ + 1) | x³ + 1 | 3x² dx |
| cos(3x) | 3x | 3 dx |
| sin³x cos x | sin x | cos x dx |
Adjust for a constant factor
If the leftover factor matches du only up to a constant, divide by that constant: with u = x², du = 2x dx, so x dx = du/2. Only constants can be adjusted this way. In ∫e^(x²) dx there is no factor of x to pair with du, and substitution does not work.
Substitute back, then check
An indefinite integral must end in terms of x: replace u with g(x). Then differentiate the answer; the chain rule should return the integrand.
Definite integrals: change the limits
For a definite integral, convert the limits too: x = a becomes u = g(a) and x = b becomes u = g(b). Then evaluate in u and never go back to x. Alternatively, substitute back first and use the original x-limits. Mixing u-limits with an x-antiderivative is a common error.
Common mistakes
- Leaving part of the integrand in x: every factor, dx included, must be rewritten in terms of u.
- Dropping a constant factor: with u = x² + 1, x dx is du/2, not du.
- Moving a variable outside the integral to force du to fit. Only constants can move; if a variable factor is left over, try a different u or another method.
- Using the old x-limits after switching to u, or u-limits after switching back.
- Forgetting to substitute back in an indefinite integral, which leaves an answer in u.
Key terms
- Integration by substitution
- A method that reverses the chain rule: when an integrand has the form f(g(x))·g′(x), set u = g(x), so du = g′(x) dx, and integrate f(u) instead. A definite integral also converts its limits to u-values.
- Chain rule
- The rule for differentiating a function inside another function: differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside.
- Antiderivative
- A function whose derivative is the given function, such as x³ for 3x². Any two antiderivatives on one interval differ by a constant, which is why answers carry + C.
- Indefinite integral
- ∫f(x) dx, the whole family of antiderivatives of f, written with + C, such as ∫2x dx = x² + C. It has no limits of integration.
- Integrand
- The function being integrated: the expression whose values contribute to the accumulation. In ∫f(x) dx, f(x) is the integrand and x is the integration variable.
Work through an example
Find ∫2x(x² + 1)⁵ dx.
Integrate 2x(x² + 1)⁵ by substitution →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
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