Advanced U-Substitution

Calculus 2 · Axiom Academy

LESSON Advanced U-Substitution Systematic techniques for when the derivative du doesn't appear on its own — algebraic rewriting and layered substitutions. 1. The Challenge: When du Isn't Obvious In basic u-substitution the derivative du sits right there in the integrand. But what about an integral like this one? The natural choice is u = x^2+1 , which gives . But the numerator hands us , not . The derivative doesn't match what's there. 2. Technique 1: Algebraic Rewriting The systematic move is to express everything in terms of u and du . Two facts unlock it: Compare what you have against what you need . Use algebra to rewrite the problem terms using u . Adjust constants and finish the substitution. Since u = x^2+1 , solving gives x^2 = u-1 . So the numerator factors and rewrites: . We still carry a stray x — but means , which mops it up. Now run the whole thing, watching each line transform into the next: 4. Technique 2: Layered Substitutions Sometimes one substitution isn't the whole job. A substitution can simplify an integral without finishing it — setting up a second technique. Nested functions: , e^ x^2 , and the like. When the first substitution simplifies but doesn't solve. When a substitution exposes a pattern that fits another technique. Every advanced u-substitution follows the same decision flow. Internalize it and the hard integrals stop looking hard. Identify the most complex part — often the innermost function. Substitute: set u equal to it and compute du .

This is the written version of the interactive lesson above. See the full Calculus 2 course.