Integration by parts

Formula

$\displaystyle \int{u}\,dv$ $\,=\,$ $uv$ $-$ $\displaystyle \int{v}\,du$

Introduction

The integrals of functions can be calculated by the integral rules but some functions are formed by the product of two or more functions. The integrals of such functions cannot be calculated by using the standard integration formulas. It requires a special integral formula to find integral of the product of two or more functions and it is popularly called the integration by parts. It is also called the partial integration.

Let $f(x)$ and $g(x)$ be two functions in terms of $x$, and denote them by two variables $u$ and $v$ respectively. In other words, $u = f(x)$ and $v = g(x)$.

Now, differentiate both functions with respect to $x$ to find their derivatives in the form of differentials.

$(1).\,\,$ $\dfrac{du}{dx}$ $\,=\,$ $f'(x)$ $\,\Longleftrightarrow\,$ $du = f'{(x)}dx$

$(2).\,\,$ $\dfrac{dv}{dx}$ $\,=\,$ $g'(x)$ $\,\Longleftrightarrow\,$ $dv = g'{(x)}dx$

The integration by parts is a mathematical process. It expresses the integral of the product of functions in terms of the integral of the product of derivative and antiderivative of the functions.

$\displaystyle \int{u}\,dv$ $\,=\,$ $uv$ $-$ $\displaystyle \int{v}\,du$

The integration by parts is mainly used when a function involved in integration and the function should be formed by the multiplication of two functions but one of them should be differentiable and other function should be integrable.

Other forms

Let us learn how to express the integration by parts formula in various forms in integral calculus.

Indefinite Integral

The integration by parts rule is also expressed mathematically in terms of $x$ as follows. It can be obtained by replacing the values of $u$ and $v$, also values of $du$ and $dv$ in the above integral equation.

$\displaystyle \int{f(x).g'(x)}\,dx$ $\,=\,$ $f(x).g(x)$ $-$ $\displaystyle \int{g(x).f'(x)}\,dx$

Definite Integral

The integration by parts law is also used in definite integration. So, the definite integration by parts property can be written as follows in the closed interval $\big[a,\,b\big]$.

$(1).\,\,$ $\displaystyle \int_{a}^{b}{u}\,dv$ $\,=\,$ $(uv)_{a}^{b}$ $-$ $\displaystyle \int_{a}^{b}{v}\,du$

$(2).\,\,$ $\displaystyle \int_{a}^{b}{f(x).g'(x)}\,dx$ $\,=\,$ $\Big(f(x).g(x)\Big)_{a}^{b}$ $-$ $\displaystyle \int_{a}^{b}{g(x).f'(x)}\,dx$

Proof

Learn how to derive the integration by parts formula mathematically in integral calculus.

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Worksheet

List of questions on integration by parts with solutions to learn how to evaluate both indefinite and definite integration by parts problems in integral calculus.

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