Integration by Substitution
Simple Explanation
Substitution rewrites a complicated integral by introducing a new variable u = g(x) for some "inner" part of the integrand — if the integrand also contains g'(x) (the derivative of that inner part), the integral simplifies dramatically in terms of u.
Why Do We Need It?
Many integrals cannot be solved with the basic rules alone — substitution is the single most common technique for reducing a complex integral to one of the standard forms already known.
Formula
The Substitution Method (u-substitution)
∫f(g(x))·g'(x) dx = ∫f(u) du, where u = g(x)
Replaces a complicated integral in x with a simpler one in a new variable u, by recognizing the integrand as the result of a Chain Rule differentiation.
- u
- — the substituted "inner function," u = g(x)
- du
- — du = g'(x) dx — the differential of u, which must also appear in the integrand
When to use it: Whenever the integrand contains a function and (a constant multiple of) that function's own derivative multiplying it.
Worked Example
Integrate using substitution
Find ∫2x(x²+1)³ dx.
Why Does This Work?
This is exactly the Chain Rule for differentiation, run in reverse: differentiating (x²+1)⁴/4 gives 4(x²+1)³/4 · 2x = 2x(x²+1)³ by the Chain Rule, which is the original integrand — so substitution is simply recognizing an integrand as the output of a Chain Rule differentiation and undoing it.
Real-Life Example
Modeling a chemical reaction rate that depends on a changing quantity
A reaction's rate might depend on the square of a concentration that is itself changing over time, creating a nested, composite expression to integrate.
Substitution lets the "inner" changing quantity be treated as a single new variable, turning a nested, hard-to-integrate expression into a simple power of u.
Practice
Find ∫2x·e^(x²) dx.
HardCommon mistake
Choosing a substitution u=g(x) when g'(x) does not actually appear (even up to a constant multiple) anywhere else in the integrand — substitution only simplifies the integral when that matching derivative is present.
Quick Review
- Choose u = g(x) for an "inner" function whose derivative also appears in the integrand.
- Rewrite the entire integral in terms of u and du, integrate, then substitute back.
- Substitution is the Chain Rule for differentiation, used in reverse.