Integration by substitution

Definition

Integration by substitution is a method used to evaluate integrals by making a substitution that simplifies the integrand. It often involves changing variables to transform the integral into a more easily solvable form.

5 Must Know Facts For Your Next Test

1. Integration by substitution is analogous to the chain rule for differentiation.
2. The goal is to rewrite the integral in terms of a new variable, usually denoted as $u$.
3. A common step is finding $du$ from $u = g(x)$ and replacing $dx$ in the original integral.
4. After performing the substitution and integrating, it is crucial to revert back to the original variable.
5. This method is especially useful when dealing with composite functions or functions within another function.

Review Questions

• Explain how integration by substitution relates to the chain rule in differentiation.
• What are the steps involved in performing an integration by substitution?
• Why is it important to convert back to the original variable after integrating?

Related terms

Chain Rule: A rule for finding the derivative of a composite function: if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

$u$-Substitution: Another term for integration by substitution where you let $u = g(x)$ and find $du$ accordingly.

Integrand: The function being integrated in an integral expression.