$x$ 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

Latest Math Problems

Email subscription

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.