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How do you integrate \[\int {{{\cot }^2}xdx} \]?

seo-qna
Last updated date: 21st Jun 2024
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Answer
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Hint: In this question, we have the trigonometric function which wants to integrate. To integrate the trigonometric function, we used a formula. And the formula is given below. First, we will convert that trigonometric function in the form of cosec function.

Complete step by step answer:
In this question, we use the word integration. First, we know about the integration. As we know Integration is defined as the process to find the function whose derivative is known. It is also called ant differentiation because the integration is the opposite process of differentiation. We know that differentiation is giving the rate of change of one variable with respect to another variable.
The main use of integration is to get back the function whose derivative is known.
If the function is \[f\left( x \right)\].then,
The integration form of that function is written as below.
\[ \Rightarrow \int_a^b {f\left( x \right)} dx\]
Here \[f\left( x \right)\] is a derivative whose upper limit is b and lower limit is a.
If
\[ \Rightarrow \dfrac{d}{{dx}}F\left( x \right) = f\left( x \right)\]
Then, we can write,
\[ \Rightarrow \int {f\left( x \right)} dx = F\left( x \right) + c\]
Here, \[F\left( x \right)\] is called the anti derivative function value of a given function\[f\left( x \right)\]. And \[c\] is known as the constant of integration and also called an arbitrary constant.
Now, come to the question. The data is given below.
\[ \Rightarrow \int {{{\cot }^2}xdx} \]
First, we convert \[\cot \]function in the form of \[\csc \]function as
\[ \Rightarrow \int {{{\cot }^2}} xdx = \int {\left( {\cos e{c^2}x - 1} \right)} dx\]
We will integrate each term separately,
\[ \Rightarrow \int {\cos e{c^2}} xdx - \int {dx} \]
Then, we integrate the above function as,
\[ \Rightarrow - {\cot }x - x + c\]

Therefore, the integration of \[\int {{{\cot }^2}xdx} \] is \[ - {\cot}x - x + c\].

Note:
As we know that if a function \[f\left( x \right)\], and integrate this function with respect to \[x\] is written as below.
\[\int {f\left( x \right)} dx\]
If \[\dfrac{d}{{dx}}f\left( x \right) = g\left( x \right)\]
Then we can write,
\[\int {g\left( x \right)} dx = f\left( x \right) + c\]
Here, \[c\]is called the constant of integration.