Integration by Parts
Integration by parts is a technique that transforms the integral of a product of two functions using the formula ∫u dv = uv - ∫v du.
Formula
\int u\, dv = uv - \int v\, du
Definition
Integration by parts is a technique for integrating a product of two functions when u-substitution does not work; the formula $\int u\, dv = uv - \int v\, du$ shifts the difficulty from one factor to the other. It is derived from the product rule: $d(uv)/dx = u\, dv/dx + v\, du/dx$, integrating both sides gives $uv = \int u\, dv + \int v\, du$, rearranged into the by-parts formula. Repeated application generates $\int u v^{(n)}\, dx = \sum_{k=0}^{n-1} (-1)^k u^{(k)} v^{(n-1-k)} + (-1)^n \int u^{(n)} v\, dx$, used to derive Taylor's theorem with integral remainder.
Example
For $\int x e^x\, dx$: choose $u = x$ and $dv = e^x\, dx$, so $du = dx$, $v = e^x$; the formula gives $xe^x - \int e^x\, dx = xe^x - e^x + C$. Some integrals need the technique applied twice and then solved algebraically: $\int e^x \cos(x)\, dx$ eventually reproduces itself on the right side, and solving gives $(e^x(\sin x + \cos x))/2 + C$. In functional analysis, integration by parts defines the adjoint of the differentiation operator: for $L^2$ functions, $\langle f', g \rangle = -\langle f, g' \rangle$ when boundary terms vanish, making $d/dx$ a skew-adjoint operator.
Key Insight
The mnemonic LIATE (Logarithm, Inverse trig, Algebraic, Trig, Exponential) suggests which factor to choose as $u$: pick the type that appears earlier in the list. When the original integral reappears on the right, collect it on one side like an algebraic equation, a "cyclic" trick that solves a whole class of integrals. The skew-adjointness of $d/dx$ under integration by parts is the mathematical reason why momentum is self-adjoint in quantum mechanics, linking calculus to the spectral theory of operators.