Integration by Parts Calculator

The Integration by Parts Calculator evaluates definite integrals of the form $$\int_a^b u \, dv$$ where $$u = ax + b$$ is a linear function and $$dv$$ is one of $$e^{cx}dx$$, $$\sin(cx)dx$$, or $$\cos(cx)dx$$. This technique, one of the most important methods in integral calculus, transforms a difficult integral into a simpler one by strategically choosing which part of the integrand to differentiate and which to integrate.

Integration by parts is derived from the product rule for differentiation. If $$u$$ and $$v$$ are differentiable functions of $$x$$, then the product rule states $$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$$. Integrating both sides and rearranging yields the integration by parts formula: $$\int u \, dv = uv - \int v \, du$$. This identity was first formulated by Brook Taylor in 1715, building on the foundational work of Newton and Leibniz.

The key insight in applying integration by parts is the choice of $$u$$ and $$dv$$. The LIATE rule provides a practical heuristic: choose $$u$$ to be the function that appears earliest in the list Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. For the integrands handled by this calculator, the linear function $$ax + b$$ is algebraic while the other factor is trigonometric or exponential, so the linear function is always chosen as $$u$$. This ensures that $$du = a \, dx$$ is simpler than $$u$$, and the remaining integral $$\int v \, du$$ can be evaluated directly.

When $$dv = e^{cx}dx$$, the antiderivative is $$v = \frac{1}{c}e^{cx}$$, and the correction integral $$\int v \, du = \frac{a}{c}\int e^{cx}dx = \frac{a}{c^2}e^{cx}$$. When $$dv = \sin(cx)dx$$, we get $$v = -\frac{1}{c}\cos(cx)$$, and $$\int v \, du = -\frac{a}{c^2}\sin(cx)$$. When $$dv = \cos(cx)dx$$, we get $$v = \frac{1}{c}\sin(cx)$$, and $$\int v \, du = \frac{a}{c^2}\cos(cx)$$. In each case, the correction integral has a known closed form, so the full result is obtained without further iteration.

Integration by parts appears throughout science and engineering. In physics, it is used to derive the equations of motion from Lagrangian mechanics, to evaluate Fourier transforms, and to compute expectation values in quantum mechanics. In electrical engineering, Laplace transforms of products rely on integration by parts. In probability theory, moments of continuous distributions often require this technique. In signal processing, the convolution of signals is closely related to iterated integration by parts.

This calculator evaluates the definite integral over a specified interval $$[a, b]$$. It displays the boundary term $$[uv]_a^b$$ and the correction term $$\left[\frac{a}{c} \cdot V(x)\right]_a^b$$ separately, allowing you to see exactly how the integration by parts formula produces the final result. Enter your parameters and bounds to get an instant, step-transparent computation.