$x$ 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.

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