AP® Calculus BC · Unit 6 · Topic 6.9
Substitution in Definite Integrals
When you substitute u for an expression in x inside a definite integral, you must also convert the bounds from x-values to the matching u-values, since the bounds are points on the x-axis, not the u-axis.
A definite integral's bounds are not just decoration — they tell you where on the x-axis to start and stop. The number under the integral sign, like x = 2, means x = 2, a specific input on the original variable's number line. The moment you substitute u = g(x), you've rebuilt the whole integral around a new variable with its own number line. The old bounds, 0 and 2, were never promises about u; they were promises about x. So you translate each one: plug x = 0 and x = 2 into u = g(x) to find out what u is at those same physical points.
This is what lets you skip the usual last step of substitution problems — swapping u back for x before plugging in numbers. If you convert the bounds to u-values up front, the entire rest of the computation, including the final plug-in, happens in u. You never need to undo the substitution, because you never left the u-world. Forget to convert the bounds, and you're stuck plugging x-numbers into a u-expression, which mixes two different scales and gives a meaningless answer, even though every algebra step looks fine.
This bound-matching habit becomes essential again when arc length integrals show up, since those integrals are built from √(1 + (dy/dx)2) dx and often need a substitution to simplify the sqrt. The same rule applies: whatever you substitute in, the bounds have to speak that same new language before you evaluate.
Key terms
- u-substitution
- A technique that rewrites an integral in terms of a new variable u to simplify the integrand.
- bounds of integration
- The two numbers on the integral sign marking the starting and ending input values for the variable being integrated.
- change of variable
- The act of replacing the integration variable and everything tied to it, including the bounds, with a new variable.
- back-substitution
- Replacing u with its original x-expression at the end of an indefinite integral, which converting the bounds lets you avoid in definite integrals.
Watch it explained
- How it worksHow does converting the limits into u values save me from substituting back at the end?
- Worked exampleFind the integral from 0 to 2 of x √(x2 + 1) dx by changing the limits into u.
- Common mistakeWhy doesn't leaving the original x limits in place after substituting still work out?
- Ties togetherHow does changing the bounds under substitution connect to the substitutions used inside arc length integrals in Unit 8?
Check yourself
End of topic.