U-Substitution with Limit Changes

Calculus 1 · Axiom Academy

EXAMPLE U-Substitution with Limit Changes Evaluate a definite integral by substituting u — and converting the bounds so you never undo the substitution Evaluate the definite integral . We'll use the substitution u=x^2 and convert the limits of integration to u — so the whole problem stays in terms of u and we never have to switch back to x . Nice work — you evaluated a definite integral with substitution by carrying the bounds along. The ideas worth keeping: Spot the inner function: the integrand had , and the x out front is (up to a constant) the derivative of the inside x^2 — the signal that u=x^2 will work. Solve for the differential: from we get , which is exactly the piece sitting in the integral. Convert the limits, don't back-substitute: running x=0 and x=2 through u=x^2 gives new bounds u=0 and u=4 . Now the integral is purely in u — no need to rewrite u back as x^2 at the end. Changing the bounds is the cleaner habit for definite integrals: once everything is in u , you evaluate immediately and the original variable never reappears.

This is the written version of the interactive lesson above. See the full Calculus 1 course.