Math Doubts

$\int \sin{x} dx$ formula Proof

$x$ is a variable and take it represents an angle of a right angled triangle. Sine function is written as $\sin{x}$ in mathematics and $dx$ is an element of integration. The integral of sin function is written in the following mathematical form.

$\int\sin{x}dx$

Derivative of cos function

Write the differentiation of cos function with respect to $x$.

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

$\implies \dfrac{d}{dx} (-\cos{x}) = \sin{x}$

Inclusion of a constant

According to differential calculus, the derivative of a constant is always zero. So, it doesn’t affect the process of differentiation. Take a constant term $C$.

$\implies \dfrac{d}{dx} (-\cos{x}+C) = \sin{x}$

Integral of sin function

The collection of all primitives of $\sin{x}$ function is called the integral of $\sin{x}$ function and is denoted by $\int \sin{x}dx$.

In this case, the primitive is $-\cos{x}$ and $C$ is called the constant of integration.

$\dfrac{d}{dx}{(-\cos{x}+C)} = \sin{x}$ $\,\Leftrightarrow\,$ $\int \sin{x}dx = -\cos{x}+C$

$\therefore \,\,\,\,\,\, \int \sin{x}dx = -\cos{x}+C$

It is called the integral of sin function in integral calculus and it is used an integration formula to deal integral of sine functions.



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