Let $f(x)$, $g(x)$, $u(x)$ and $v(x)$ represent four functions in $x$, and also $c_1$ and $c_2$ are constants. Now, let’s derive the formula for integral sum rule in mathematical form by the relation between differentiation and integration in integral calculus.

Add a constant $c_1$ to the function $u(x)$, then the sum of them can be written as expression $u(x)+c_1$. Let $f(x)$ be the derivative of the mathematical expression $u(x)+c_1$. The relationship between them can be written in mathematical form by the differentiation.

$\dfrac{d}{dx}{\, \Big(u(x)+c_1\Big)} \,=\, f(x)$

According to the integral calculus, the function $u(x)$ is called a primitive or anti-derivative of function $f(x)$ and $c_1$ is called the constant of integration. It can be written in mathematical form by the indefinite integration.

$\implies$ $\displaystyle \int{f(x) \,}dx \,=\, u(x)+c_1$

$\,\,\, \therefore \,\,\,\,\,\,$ $u(x)+c_1 \,=\, \displaystyle \int{f(x) \,}dx$

Similarly, add a constant $c_2$ to the function $v(x)$. Now, the addition of them is written as an expression $v(x)+c_2$. Assume, the derivative of expression $v(x)+c_2$ is equal to $g(x)$. It can be written mathematically by the differential calculus.

$\dfrac{d}{dx}{\, \Big(v(x)+c_2\Big)} \,=\, g(x)$

As per integral calculus, the function $v(x)$ is called the integral of $g(x)$ with respect to $x$ and $c_2$ is called a constant of integration. The relationship between them can be written mathematically by the integration.

$\implies$ $\displaystyle \int{g(x) \,}dx \,=\, v(x)+c_2$

$\,\,\, \therefore \,\,\,\,\,\,$ $v(x)+c_1 \,=\, \displaystyle \int{g(x) \,}dx$

Now, add both mathematical expressions and evaluate its derivative.

$\implies$ $\dfrac{d}{dx}{\, \Big[\Big(u(x)+c_1+v(x)+c_2\Big)\Big]}$ $\,=\,$ $\dfrac{d}{dx}{\, \Big[\Big(u(x)+c_1\Big)+\Big(v(x)+c_2\Big)\Big]}$

The derivative of sum is equal to sum of their derivatives as per sum rule of differentiation.

$\implies$ $\dfrac{d}{dx}{\, \Big[\Big(u(x)+c_1+v(x)+c_2\Big)\Big]}$ $\,=\,$ $\dfrac{d}{dx}{\,\Big(u(x)+c_1\Big)}$ $+$ $\dfrac{d}{dx}{\, \Big(v(x)+c_2\Big)}$

In the first step, we have assumed that the derivatives of the expressions $u(x)+c_1$ and $v(x)+c_2$ are $f(x)$ and $g(x)$ respectively.

$\implies$ $\dfrac{d}{dx}{\, \Big[\Big(u(x)+c_1+v(x)+c_2\Big)\Big]}$ $\,=\,$ $f(x)+g(x)$

As per the relationship between differentiation and integration, the mathematical expression in differential form can be written integral form.

$\implies$ $\displaystyle \int{\Big(f(x)+g(x)\Big)\,}dx$ $\,=\,$ $u(x)+c_1$ $+$ $v(x)+c_2$

In first step, we have derived that

$(1)\,\,\,$ $u(x)+c_1 \,=\, \displaystyle \int{f(x)\,}dx$

$(2)\,\,\,$ $v(x)+c_2 \,=\, \displaystyle \int{g(x)\,}dx$

Now, replace the values of expressions $u(x)+c_1$ and $v(x)+c_2$ in the above equation.

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \int{\Big(f(x)+g(x)\Big)\,}dx$ $\,=\,$ $\displaystyle \int{f(x) \,}dx$ $+$ $\displaystyle \int{g(x) \,}dx$

Therefore, it is proved that the indefinite integral of sum of functions is equal to sum of their integrals, and it is called as the sum rule of integration.

Latest Math Topics

May 21, 2023

May 16, 2023

May 10, 2023

May 03, 2023

Latest Math Problems

May 09, 2023

A best free mathematics education website that helps students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved