Integration by Parts
Calculus & Advanced MathIntegration 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 ∫u dv = uv - ∫v du shifts the difficulty from one factor to the other.
Example
$\int x e^x\, dx$: choose $u = x$ and $dv = e^x\, dx$. Then $du = dx$, $v = e^x$. Formula gives: $xe^x - \int e^x\, dx = xe^x - e^x + C$.
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.
Definition
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 as $\int u\, dv = uv - \int v\, du$. Sometimes must be applied repeatedly or used to solve for the original integral when it appears on the right side.
Example
$\int e^x \cos(x)\, dx$: apply by parts twice. First: $e^x\sin(x) - \int e^x\sin(x)\, dx$. Second application produces $\int e^x\cos(x)\, dx$ again. Solving: $\int e^x\cos(x)\, dx = (e^x(\sin x + \cos x))/2 + C$.
Key Insight
When the original integral reappears on the right, collect it on one side like an algebraic equation. This "cyclic" trick solves a whole class of integrals.
Definition
Integration by parts is the integral analogue of the product rule (Leibniz rule). The formula extends to: $\int_a^b u v'\, dx = [uv]_a^b - \int_a^b v u'\, dx$. Repeated application generates the integration-by-parts formula: $\int u v^{(n)}\, dx = \sum_{k=0}^{n-1} (-1)^k u^{(k)} v^{(n-1-k)} + (-1)^n \int u^{(n)} v\, dx$, used to derive Taylor's theorem with integral remainder.
Example
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 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.