The product rule of differentiation—d(uv)/dx = u·dv/dx + v·du/dx—has a corresponding technique for integration. Rearranged and integrated, it becomes the integration by parts formula: ∫u dv = uv − ∫v du. This single identity unlocks integrals of products that no other technique can handle, from ∫x·eˣ dx to ∫ln(x) dx to the integral definitions of the Gamma function and Laplace transforms. Integration by parts is calculus's most powerful and versatile integration technique.
Deriving the Formula
Starting from the product rule: d(uv) = u dv + v du. Integrating both sides: ∫d(uv) = ∫u dv + ∫v du, giving uv = ∫u dv + ∫v du. Rearranging: ∫u dv = uv − ∫v du. The art of applying this formula lies in choosing which part of the integrand to call u (to differentiate) and which to call dv (to integrate). A good choice makes the remaining integral ∫v du simpler than the original; a poor choice makes it harder or sends you in circles.
The LIATE Rule
The mnemonic LIATE suggests which function to choose as u: Logarithms, Inverse trig, Algebraic (polynomials), Trigonometric, Exponential—in that order of preference. This works because differentiating logarithms and inverse trig simplifies them dramatically, while exponentials and trig functions cycle when differentiated or integrated. For ∫x·eˣ dx: choose u = x (algebraic) and dv = eˣ dx. Then du = dx and v = eˣ, giving ∫x·eˣ dx = x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C = eˣ(x − 1) + C.
Repeated Application
Some integrals require integration by parts multiple times. For ∫x²·eˣ dx, applying the formula once gives 2∫x·eˣ dx, which requires a second application. The tabular method (also called the DI method) organizes repeated applications: create two columns, alternately differentiating u and integrating dv, then multiply diagonally with alternating signs. This method efficiently handles integrals like ∫xⁿ·eˣ dx for any polynomial xⁿ without laboriously applying the formula n times from scratch.
The Circular Case
For ∫eˣ·sin(x) dx, integration by parts leads to another integral of the same form. Applying the formula twice: ∫eˣ·sin(x) dx = eˣ·sin(x) − eˣ·cos(x) − ∫eˣ·sin(x) dx. Treating the repeated integral as an unknown and solving algebraically gives the answer. This circular application appears frequently with products of exponentials and trigonometric functions, and the algebraic solution technique is elegant—the integral solves itself through an equation rather than a direct antiderivative.
Applications in Physics and Engineering
Integration by parts is ubiquitous in applied mathematics. The Laplace transform—essential for control systems and differential equations—is defined as an integral that frequently requires integration by parts to evaluate. Quantum mechanics uses it to derive properties of operators: showing that a differential operator is Hermitian (self-adjoint) requires integrating by parts and verifying that boundary terms vanish. In electrostatics, integrating the gradient of a potential by parts connects volume integrals to surface integrals, yielding Green's theorems that underpin electrostatic potential theory.
Conclusion
Integration by parts is the integration counterpart of the product rule—a systematic technique for reducing integrals of products to (hopefully) simpler integrals. Its elegance lies in trading a hard integral for an easier one through a single algebraic identity. From computing Laplace transforms to proving mathematical physics identities to simply integrating x·eˣ, integration by parts is an indispensable tool that every calculus student must master and every applied mathematician relies on regularly.