U-Substitution
Calculus & Advanced MathU-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. It is the integration version of the chain rule.
Example
To integrate $\int 2x (x^2 + 1)^5\, dx$, let $u = x^2 + 1$, so $du = 2x\, dx$. The integral becomes $\int u^5\, du = u^6/6 + C = (x^2+1)^6/6 + C$.
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.
Definition
Let $u = g(x)$, then $du = g'(x)\, dx$. The substitution transforms $\int f(g(x))g'(x)\, dx$ into $\int f(u)\, du$. For definite integrals, also change the limits: if $u = g(x)$, the new limits are $g(a)$ and $g(b)$.
Example
$\int_0^1 x e^{x^2}\, dx$: let $u = x^2$, $du = 2x\, dx$, so $x\, dx = du/2$. New limits: $u(0)=0$, $u(1)=1$. Integral $= (1/2)\int_0^1 e^u\, du = (1/2)(e - 1)$.
Key Insight
Changing limits when substituting in a definite integral saves you from having to substitute back at the end, reducing errors.
Definition
U-substitution is the integration form of the chain rule, formalized as the change-of-variables theorem: if $g$ is continuously differentiable on $[a,b]$ and $f$ is continuous on $g([a,b])$, then $\int_a^b f(g(x))g'(x)\, dx = \int_{g(a)}^{g(b)} f(u)\, du$. In multiple dimensions, this generalizes to the Jacobian change-of-variables formula.
Example
In multivariable calculus: converting $\iint$ 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 Jacobian determinant is the multidimensional analogue of $du/dx$, measuring how area (or volume) scales under the coordinate change. This is essential in probability (change-of-variables for random variables) and differential geometry.