How do the limits change when using u-substitution in a definite integral?

Conditions

• The integral is a definite integral.
• A substitution $u = u(x)$ is being used.
• The original limits of integration are given in terms of $x$.

Reasoning, step by step

1. Identify the original lower and upper limits of integration in terms of $x$.
2. Write down the substitution equation relating $u$ to $x$.
3. Substitute the original lower $x$-limit into the equation to find the new lower $u$-limit.
4. Substitute the original upper $x$-limit into the equation to find the new upper $u$-limit.
5. Rewrite the definite integral using the new $u$-limits and the transformed integrand.

Example

In the worked example, the instructor states, "And then we just have to change the bounds of integration." He substitutes $x=0$ to get $u=1$, and $x=32/9$ to get $u=9$, converting the integral bounds from $[0, 32/9]$ in $x$ to $[1, 9]$ in $u$.

Common misconceptions

• Forgetting to change the bounds of integration when switching from $dx$ to $du$.
• Substituting the $x$-limits into the wrong equation or failing to evaluate the arithmetic correctly.
• Evaluating the definite integral using the original $x$-limits after transforming the integrand to $u$.

Watch the explanation

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