Integration By Parts 26

1. Core Concept Overview & Intuition

Integration by parts is a technique for integrating products of functions. It is the integration counterpart of the product rule for differentiation. The central idea is to transform an integral that is difficult to evaluate directly into a simpler integral that can be handled using the basic formulae and the extension formulae already established.

The method rests on a single identity. If $u$ and $v$ are differentiable functions of $x$, then the product rule gives the derivative of $uv$. Reversing that differentiation process yields a formula that expresses the integral of $u$ times the derivative of $v$ in terms of $uv$ and the integral of $v$ times the derivative of $u$. The practical consequence is that the original integrand is split into two factors: one factor is differentiated, and the other is integrated. A judicious choice of which factor to differentiate and which to integrate reduces the problem to a standard integral.

The technique is especially valuable when the integrand is a product of unlike types of functions, such as an algebraic function multiplied by a logarithmic function, or an algebraic function multiplied by an exponential or trigonometric function. It also provides the standard route to the integral of the natural logarithm, which does not appear among the basic formulae.

The intuition is one of trading a hard integral for an easier one. Each application of the formula replaces the integral of $u\,dv$ by the boundary term $uv$ together with the integral of $v\,du$. Success depends on choosing the factors so that $v\,du$ is simpler than $u\,dv$. When the integrand is a single logarithmic function, the device is to regard the integrand as the product of that logarithm with the constant function $1$, so that the formula applies.

2. Mathematical Formalization & Core Definitions

> Definition (Integration by Parts). Integration by parts is an integration technique that rewrites the integral of a product of two functions in terms of the product of the two functions and the integral of their reversed product. It is derived from the product rule of differentiation and is used when the integrand is a product whose direct integration is not immediate.

> Definition (Integrand). In an integral written $\int f(x)\,dx$, the function $f(x)$ is called the integrand of the integral. The integral sign cannot be divorced from $dx$ when the integral is with respect to $x$.

> Definition (Constant of Integration). Since any additive constant disappears under differentiation, a constant $c$ is always appended after the process of integration. Thus an indefinite integral is determined only up to an additive constant.

The following basic formulae from earlier work are used repeatedly in this lesson:

$$
\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c,\qquad n\neq -1
$$

$$
\int \frac{1}{x}\,dx=\ln x+c
$$

$$
\int e^{x}\,dx=e^{x}+c
$$

$$
\int a^{x}\,dx=\frac{a^{x}}{\ln a}+c,\qquad a>0
$$

$$
\int \cos x\,dx=\sin x+c
$$

$$
\int \sin x\,dx=-\cos x+c
$$

$$
\int \sec^{2}x\,dx=\tan x+c
$$

$$
\int \sec x\tan x\,dx=\sec x+c
$$

$$
\int \operatorname{cosec}x\cot x\,dx=-\operatorname{cosec}x+c
$$

Here $x$ is the variable of integration, $n$ is a real exponent with $n\neq -1$, $a$ is a positive constant base, $e$ is the base of natural logarithms, and $c$ is the constant of integration.