U-Substitution

Business Calculus · Axiom Academy

The chain rule in reverse - a powerful integration technique Remember the chain rule for derivatives? U-substitution reverses this process for integration: If you see f'(g(x)) · g'(x), you can substitute u = g(x) There's a "inner function" and its derivative (or a constant multiple) The integrand is a composition of functions Basic rules don't directly apply Identify u: Choose the "inner function" - usually what's inside parentheses, under a root, or in an exponent Find du: Differentiate u with respect to x: du = u'(x)\,dx Substitute: Replace all x's with u's and dx with du Integrate: Perform the (now simpler) integral in terms of u Back-substitute: Replace u with the original expression in x 3 Example 1: Power of a Linear Function 4 Example 2: Exponential with Linear Exponent Notice that 3x² is the derivative of x³. This is the telltale sign that u-substitution will work! Pro Tip: Adjusting for Constants If you're "missing" a constant multiple, you can adjust: Multiply and divide to create the du you need.

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