Calculus 1 — 26.5: U-Substitution in Definite Integrals скачать в хорошем качестве

Calculus 1 — 26.5: U-Substitution in Definite Integrals

Two methods for handling bounds in u-substitution: keep the original x-bounds and back-substitute, or convert the bounds to u at the start. Both approaches are worked through step by step on the integral of 4x(x²−1)³ from 0 to 2, showing why each yields 40—and what goes wrong when you mix variables with the wrong bounds. Two additional examples cover trig substitution with sin⁵(2x)cos(2x) and the technique of solving for x when leftover terms remain after substituting. Key concepts covered: • Setting up u-substitution: choosing u, computing du, canceling terms • Method 1: integrate in u, back-substitute to x, then evaluate at original x-bounds • Method 2: convert bounds from x to u at the start, evaluate directly in u • Why mixing u-expressions with x-bounds produces wrong answers • Chain rule factor in trig substitution: u = sin(2x), du = 2cos(2x) dx • Handling leftover x-terms by solving u = g(x) for x and substituting • Distributing before integrating products of polynomials in u • Exponent arithmetic: computing (√2/2)⁶ step by step • A complete decision flowchart for u-substitution in any integral ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 4.5:  The Fundamental T...