AP® Calculus AB · 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 u-values, so the whole problem finishes in terms of u with no need to switch back.
A definite integral's bounds are values of x — they tell you where on the x-axis to start and stop. When you substitute u = g(x), you're changing variables entirely, so a bound like x = √π is meaningless in u-world until you plug it into g to get the matching u-value. Skipping this step is the single biggest substitution error: the numbers 0 and √π don't just carry over, they transform.
Once you convert both bounds through u = g(x), you can evaluate the antiderivative in u and plug in the new bounds directly — no need to undo the substitution and rewrite everything back in x. That's the whole payoff: converting bounds early saves you a step at the end, because the Fundamental Theorem of Calculus lets you evaluate F(u) at the u-bounds and get your final number immediately.
This sits right after indefinite-integral substitution, where you had to resubstitute back to x because there were no bounds to convert. In definite integrals, bounds give you a shortcut: find du, rewrite the integral entirely in u, swap in the new limits, and finish. It also connects forward to more complex accumulation and area problems, where misordered or flipped bounds (new upper bound smaller than new lower bound) still work correctly as long as you trust the algebra and don't force the bounds back into increasing order by hand.
Key terms
- u-substitution
- A technique that rewrites an integral in terms of a new variable u to make it easier to antidifferentiate.
- bounds of integration
- The starting and ending values between which a definite integral is evaluated.
- Fundamental Theorem of Calculus
- The rule that a definite integral equals the antiderivative evaluated at the upper bound minus at the lower bound.
- du
- The differential of u, found by differentiating the substitution equation u = g(x) with respect to x.
Watch it explained
- How it worksHow does changing the bounds save you from converting back to x at the end?
- Worked exampleFind the definite integral of 2x cos(x2) dx from 0 to √π by changing the bounds.
- Common mistakeWhy isn't it correct to keep the original bounds after substituting?
- Going furtherWhat happens when a substitution makes the new upper bound smaller than the lower one?
Check yourself
End of topic.