Math Doubts

Proof of Integral of cosx formula

$x$ 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.

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

Derivative of sin function

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

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

Inclusion of an Arbitrary constant

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 $\sin{x}$.

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

Integral of cos function

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.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved