u-Substitution

u-Substitution: idea

Use u-substitution when an integrand contains a function and, alongside it, a factor that looks like its derivative. The goal is to replace a complicated inner expression with a new variable u, turning the integral into a simpler one.

Method

1. Choose the inner expression: Pick the part of the integrand that becomes simpler after substitution. A good choice is often inside parentheses, a power, a root, or an exponential.
2. Set u equal to that expression: Write u = ....
3. Differentiate both sides: Compute du so you can replace the matching factor in the integral.
4. Rewrite the entire integral in terms of u: Replace every occurrence of the inner expression and the differential.
5. Integrate in u: The new integral should be simpler.
6. Substitute back: Replace u with the original expression and simplify the result.

Check

Differentiate your final answer to see whether you recover the original integrand. If the derivative matches after simplification, the substitution was done correctly.

Common pitfalls

• Forgetting to replace the differential.
• Choosing u so that du does not match any factor in the integral.
• Stopping before substituting back to the original variable.

Goal

Your final answer should be exact and fully simplified.