Math Doubts

Proof of Integral of sinx formula

$x$ is a variable, which is considered as an angle of a right triangle and the sine function is written as $\sin{x}$ in trigonometric mathematics. The indefinite integral of $\sin{x}$ with respect to $x$ is written as follows to find the integration of sine function in calculus.

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

Derivative of cos function

Write the derivative of cos function with respect to $x$ formula for expressing the differentiation of cosine function in mathematical form.

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

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

Inclusion of an Arbitrary constant

According to differential calculus, the derivative of a constant is always zero. So, it doesn’t affect the process of the differentiation if an arbitrary constant $(c)$ is added to the trigonometric function $-\cos{x}$.

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

Integral of sin function

The collection of all primitives of $\sin{x}$ function is called the indefinite integral of $\sin{x}$ function, which is written in the following mathematical form in integral calculus.

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

In this case, the primitive or an antiderivative of $\sin{x}$ is $-\cos{x}$ and the constant of integration $c$.

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

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

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

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

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.

Learn more