Key takeaway: U-substitution is an integration technique that renames an inner expression as u, rewrites the integral entirely in terms of u and du, and reverses the chain rule.
Grade 12 – AP Calculus ABAP Calculus AB Unit 6Calculus
Definition of U-Substitution
U-substitution simplifies an integral that contains a function inside another function, together with a multiple of the inner function’s derivative. Renaming the inner part as \(u\) turns a tangled expression into a basic one.
The technique is the chain rule run backward. The key is to look for a pair: an inside piece and, somewhere else in the integrand, its derivative up to a constant. If that pair is present, every \(x\) can be traded for \(u\).
Formula and notation
Choose \(u=q(x)\) and compute \(du=q^{\prime}(x)\,dx\). Then
\[ \int s\big(q(x)\big)\,q^{\prime}(x)\,dx=\int s(u)\,du. \]
After integrating in \(u\), substitute back. For a definite integral, the limits can be converted to \(u\)-values instead.
Key facts
- Pick u to be the inside of a composite function, or a denominator or radicand.
- Every x and dx must disappear; the new integral may contain only u and du.
- Return to the original variable at the end, unless the limits were converted.
- U-substitution fails when no inner piece has its derivative nearby; try another method then.
U-Substitution example
Evaluate \(\int 8x\,(3x^2+1)^5\,dx\).
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Let \(u=3x^2+1\). Then \(du=6x\,dx\), so \(8x\,dx=\frac{8}{6}\,du=\frac{4}{3}\,du\).
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Rewrite: \(\frac{4}{3}\int u^5\,du=\frac{4}{3}\cdot\frac{u^6}{6}+C=\frac{2}{9}u^6+C\).
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Substitute back \(u=3x^2+1\).
The integral equals \(\frac{2}{9}(3x^2+1)^6+C\).
Common mistake
Watch out
A student sets \(u=3x^2+1\) but writes \(\int 8x\,u^5\,du\), leaving an \(x\) inside an integral in \(u\).
Why it is wrong
After a substitution, the integrand and the differential must both be in terms of \(u\). The factor \(8x\,dx\) has to be replaced by \(\frac{4}{3}\,du\). If an \(x\) remains, either the substitution was a poor choice or the conversion is unfinished. Good u-substitution leaves nothing but u behind.
Related terms and lessons
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