$x$ is a variable, which represents an angle of a right triangle and the cosine function is written as $\cos{x}$ in trigonometry. The indefinite integral of $\cos{x}$ with respect to $x$ is mathematically written in the following mathematical form.

$\displaystyle \int{\cos{x} \,}dx$

Write the derivative of sin function with respect to $x$ formula for writing the differentiation of sine function in mathematical form.

$\dfrac{d}{dx}{\, \sin{x}} \,=\, \cos{x}$

As per differential calculus, the derivative of a constant is always zero. So, it does not change the differentiation even an arbitrary constant ($c$) is added to the trigonometric function $\sin{x}$.

$\implies$ $\dfrac{d}{dx}{(\sin{x}+c)} \,=\, \cos{x}$

According to integral calculus, the collection of all primitives of $\cos{x}$ function is called the indefinite integral of $\cos{x}$ function and it can be expressed in the following mathematical form.

$\displaystyle \int{\cos{x} \,}dx$

Here, the primitive or an antiderivative of $\cos{x}$ function is $\sin{x}$ and the constant of integration $c$.

$\dfrac{d}{dx}{(\sin{x}+c)} = \cos{x}$ $\,\Longleftrightarrow\,$ $\displaystyle \int{\cos{x} \,}dx = \sin{x}+c$

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{\cos{x} \,}dx = \sin{x}+c$

Therefore, it has proved that the indefinite integral or antiderivative of cosine function is equal to the sum of the sine function and the constant of integration.

Latest Math Topics

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Jul 29, 2022

Jul 17, 2022

Jun 02, 2022

Apr 06, 2022

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

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

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved