Integration by substitution

Definition

Integration by substitution is a method for finding integrals by making a substitution to simplify the integral. It involves changing variables to rewrite an integral in a simpler form.

5 Must Know Facts For Your Next Test

1. The substitution method often uses $u$-substitution, where you let $u = g(x)$ and then find $du = g'(x)dx$.
2. To apply this method, the function inside the integral must be written in terms of $u$ and $du$ after the substitution.
3. Reversing the substitution at the end of the process is necessary to return to the original variable.
4. Integration by substitution can simplify integrals involving composite functions or products of functions.
5. It's essential to correctly identify and substitute both $u$ and its derivative $du$ for successful integration.

Review Questions

• How do you determine what to substitute for $u$ in integration by substitution?
• What steps are involved in reversing the substitution after integrating?
• Explain why integration by substitution is useful for solving complex integrals.

Related terms

Definite Integral: An integral that computes the area under a curve between two specific limits. Represented as $\int_{a}^{b} f(x)dx$.

$u$-Substitution: A technique used in integration where one sets $u = g(x)$, simplifying an integral into a new variable form.

Change of Variables: A broader mathematical concept that includes techniques like integration by substitution to simplify problems involving different variables.