U-Substitution

Calculus 1 · Axiom Academy

LESSON U-Substitution for Definite Integrals A substitution doesn't just rename a variable — it re-draws the area in a new world. See why, and meet the two ways to evaluate. 1. The Substitution Reshapes the Area Our integral is a perfect candidate for u -substitution. Let u = x^2+1 . Then , so the stray sitting in the integral is exactly half of du : Set the substitution and differentiate The integral becomes a clean 1/u in the new variable 2. Method 1: Substitute Back to x The traditional route: integrate in u , then convert the antiderivative back to x before evaluating at the original limits x=0 and x=1 . 3. Method 2: Change the Limits The cleaner route: when you swap x for u , send the limits along too. Feed each x -limit through u=x^2+1 and you never have to come back to x at all. Rewrite with the new limits: . Evaluate at u=1 and u=2 — no substituting back. Familiar from indefinite integrals Lets you check the antiderivative in x But: an extra conversion step, more room for slips Just remember to convert both limits Everything above is one theorem. Whenever u=g(x) is smooth, the substitution carries the limits with it and the two integrals are equal as areas : The rule in words: when x=a , the lower limit becomes u=g(a) ; when x=b , the upper limit becomes u=g(b) . Then integrate and evaluate entirely in u . You've seen u -substitution as an area-preserving change of variable, and you know both ways to finish a definite integral. Scroll up to revisit any step.

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