Introduction
U-substitution is one of the most important integration techniques on the AP Calculus exam. It is often the first method students learn after basic antiderivatives, and it shows up repeatedly in both AP Calculus AB and BC questions. When applied correctly, u-substitution simplifies complicated integrals into forms you can solve easily.
In this guide, we’ll cover what u-substitution is, when to use it, step-by-step instructions, common errors, and examples tailored for AP exam success.
What is U-Substitution?
U-substitution is a method used to simplify integrals by substituting part of the integrand with a new variable, typically u. It works especially well when the integral involves a composite function.
The basic process:
- Identify an inner function that you can set equal to u.
- Differentiate u to find du.
- Rewrite the integral entirely in terms of u.
- Solve the integral.
- Substitute back the original variable.
When to Use U-Substitution
You should consider u-substitution when:
- The integrand contains a function and its derivative.
- You see something like f(g(x))·g’(x), which is a sign substitution will work.
- The derivative of the inside function is either present directly or is a simple multiple of what’s in the integrand.
Step-by-Step Example 1
Solve: ∫2x cos(x²) dx
- Let u = x² → du = 2x dx
- Rewrite integral: ∫cos(u) du
- Integrate: sin(u) + C
