Method guideCalculus IFoundational9 min read

u-substitution as the reverse Chain Rule

Replace a repeated inner expression and its derivative with one variable so the antiderivative’s structure becomes visible.

u=g(x),\qquad du=g'(x)\,dx

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Reviewed July 11, 2026

Recognition testStart here

Look for a composite expression g(x) together with a constant multiple of g′(x). If both are present, substitution is usually the clean first move.

What the substitution accomplishes

The method is not a letter swap. It packages an inside function and its differential so the integral becomes a familiar basic form.

A successful substitution removes every x. If x remains, either rewrite it using u or choose a different substitution.

Constant multiples are harmless

The derivative does not need to appear exactly. If it differs by a nonzero constant, factor that constant in or out. What matters is structural proportionality.

Do not invent a missing variable factor; constants can be adjusted, variable expressions cannot.

Definite integrals have two clean options

Either find an antiderivative in u and return to x before using the original bounds, or convert the bounds to u and stay in u. Mixing x-bounds with a u-integrand is inconsistent.

Worked exampleReverse the Chain Rule

Evaluate ∫ 2x cos(x² + 1) dx.

u=x^2+1

Choose the repeated inner expression.

du=2x\,dx

Its differential matches the remaining factor.

\int\cos u\,du=\sin u+C

Integrate the basic form.

Result

\boxed{\sin(x^2+1)+C}

Watch for

Common mistakes

Changing part of the integrand to u while leaving unexplained x-terms.
Forgetting to substitute back in an indefinite integral.
Using original x-bounds after converting the integrand to u.

Keep

Three takeaways

Find an inside function and its derivative.
Adjust constants, not variable factors.
A complete substitution uses one variable at a time.