$x$ is a variable and also denotes an angle of a right triangle. The cosecant and cotangent functions are written as $\csc{x}$ or $\operatorname{cosec}{x}$ and $\cot{x}$ respectively. The indefinite integration of product of cosecant and cot functions with respect to $x$ is written in the following mathematical form in integral calculus.

$\displaystyle \int{\csc{x}\cot{x} \,}dx$

Now, let’s derive the proof for the indefinite integration of product of $\csc{x}$ and $\cot{x}$ functions with respect to $x$ in integral calculus.

It is proved that the derivative of cosecant function is equal to the negative of product of cosecant and cotangent functions.

$\dfrac{d}{dx}{\, \csc{x}} \,=\, -\csc{x}\cot{x}$

In fact, the derivative of a constant is zero. So, the derivative of cosecant function is same even though a constant is added to the cosecant function in differentiation. Add an arbitrary constant to cosecant function and now, differentiate the trigonometric function with respect to $x$.

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

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

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

$\implies$ $\dfrac{d}{dx}{(-\csc{x}-c)} \,=\, \csc{x}\cot{x}$

$\implies$ $\dfrac{d}{dx}{(-\csc{x}+c_{1})} \,=\, \csc{x}\cot{x}$

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

The collection of all primitives of product of $\csc{x}$ and $\cot{x}$ function is called the integration of product of cosecant and cot functions. As per the integral calculus, It can be written in mathematical form as follows.

$\displaystyle \int{\csc{x}\cot{x} \,}dx$

The primitive or an antiderivative of $\csc{x}\cot{x}$ function is equal to the sum of the $-\csc{x}$ function and the constant of integration ($c$).

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

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{\csc{x}\cot{x} \,}dx = -\csc{x}+c$

Therefore, it is proved that the indefinite integration of product of cosecant and cot functions with respect to a variable is equal to the sum of the negative cosecant function and integral constant.

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.