Definition: U-substitution simplifies an integral by replacing an inner function with a new variable u, reversing the chain rule so the integral becomes a basic one.
When does it work?
Look for an inner function whose derivative (up to a constant) also appears in the integrand. Let u be that inner function, find du, rewrite everything in terms of u, integrate, then substitute back.
How do you carry it out?
Worked example: ∫ 12x2(x3 + 4)3 dx. Let u = x3 + 4, so du = 3x2 dx. Then 12x2 dx = 4 du.
The integral becomes ∫ 4u3 du = u4 + C = (x3 + 4)4 + C.
Check: the derivative is 4(x3 + 4)3 times 3x2 = 12x2(x3 + 4)3.
What do students forget?
For a definite integral, either change the limits to u-values or switch back to x before using the original limits. Mathbot also checks that no leftover x remains in the integrand.
Example: ∫ 12x^2 (x^3 + 4)^3 dx = (x^3 + 4)^4 + C