Question

How do you find the integral of \[\cot {{x}^{5}}\csc {{x}^{2}}\] ?

Answer 1

Hint:In the given question, we have been asked to integrate the given constant. In order to solve the question, we integrate the numerical by using the basic concept of integration. First we need to take out the constant part out of the integration. Later we will need to integrate the variable part using a suitable integration formula and we will get our required answer.

Complete step by step answer:
We have given,
\[\Rightarrow \int{\cot {{x}^{5}}\csc {{x}^{2}}dx}\]
Let I be the integration of the given equation.
Therefore, we can write the integration as,
\[\Rightarrow I=\int{\cot {{x}^{5}}\csc {{x}^{2}}dx}\]
Substituting cot(x) = t in the above expression;
\[\Rightarrow \cot x=t\Rightarrow \dfrac{d}{dt}\cot x=\dfrac{dt}{dt}\Rightarrow -{{\csc }^{2}}x\dfrac{dx}{dt}=1\Rightarrow {{\csc }^{2}}xdx=-dt\]
Therefore,
Substituting\[{{\csc }^{2}}xdx=-dt\], we have
\[\Rightarrow I=\int{{{t}^{5}}dt}\]
Using the integration formula, i.e.
\[\Rightarrow \int{{{t}^{n}}dt=\dfrac{{{t}^{n+1}}}{n+1}}+c\], where n is not equal to -1,
Thus applying this formula, we have
\[\Rightarrow I=\dfrac{{{t}^{6}}}{6}+C\]
Undo the substitution in the above expression, we have obtained
\[\Rightarrow I=\dfrac{{{\cot }^{6}}x}{6}+C\]
\[\therefore \int{\cot {{x}^{5}}\csc {{x}^{2}}dx}=\dfrac{{{\cot }^{6}}x}{6}+C\]

Hence,the integral of \[\cot {{x}^{5}}\csc {{x}^{2}}\] is $\dfrac{{{\cot }^{6}}x}{6}+C$.

Note:To solve any given numerical, or function, there are different types of integration methods like integration by substitution, integration by parts, etc. Here we need to remember that we have to put the constant term after the integration. we should know all the basic methods to integrate the given functions. There are other methods for integration. These are integration by-parts and integration by partial fractions. It is useful to know all the methods for integration so that we can choose one for computation according to the convenience and ease of calculation. We should do all the calculations carefully and explicitly to avoid making errors. Integration is the part of calculus that includes the differentiation. Integration refers to the, add up smaller parts of any area given, volume given, etc to represents the whole value.