U-Substitution
U-substitution is an integration technique that simplifies a complex integral by replacing an inner expression with the variable u.
Formula
\int f(g(x))g'(x)\, dx = \int f(u)\, du, \, u=g(x)
Definition
U-substitution is a trick for integration where you rename a complicated inner piece of the function as "$u$" to make the integral simpler, the integration version of the chain rule. Let $u = g(x)$, so $du = g'(x)\, dx$; this transforms $\int f(g(x))g'(x)\, dx$ into $\int f(u)\, du$, and for definite integrals you also change the limits: if $u = g(x)$, the new limits become $g(a)$ and $g(b)$. Formalized as the change-of-variables theorem, this generalizes in multiple dimensions to the Jacobian change-of-variables formula.
Example
To integrate $\int 2x (x^2 + 1)^5\, dx$, let $u = x^2 + 1$, so $du = 2x\, dx$; the integral becomes $\int u^5\, du = u^6/6 + C = (x^2+1)^6/6 + C$. For $\int_0^1 x e^{x^2}\, dx$: let $u = x^2$, $du = 2x\, dx$, and the limits become $u(0)=0$, $u(1)=1$, giving $(1/2)\int_0^1 e^u\, du = (1/2)(e - 1)$. In multivariable calculus, converting a double integral over a disk to polar coordinates uses the Jacobian $|J| = r$: $\iint f(x,y)\, dA = \iint f(r\cos\theta, r\sin\theta) \, r \, dr\, d\theta$.
Key Insight
The key is spotting an expression whose derivative also appears in the integral; when you see a function and its derivative together, u-substitution will work, and changing limits in a definite integral saves you from substituting back at the end. The Jacobian determinant is the multidimensional analogue of $du/dx$, measuring how area (or volume) scales under a coordinate change, essential in probability (change-of-variables for random variables) and differential geometry.