Answer
384.3k+ views
Hint: Here we have to find the integral of a given function. We will use the trigonometric ratios and try to write the given ratio in standard integral form. Here first we will use the substitution method. Let’s put \[u = \cot x\] . Because on further we can write \[\cot x = \dfrac{{\cos x}}{{\sin x}}\] on taking differentiation with the help of product and quotient rule will help in solving the problem in lesser steps and the best approach.
Complete step by step solution:
Given that find the integral of \[\dfrac{1}{{{{\sin }^2}x}}\]
So that is \[\int {\dfrac{1}{{{{\sin }^2}x}}dx} \]
Now we will substitute \[u = \cot x\]
On writing \[\cot x = \dfrac{{\cos x}}{{\sin x}}\]
Next we will differentiate the above terms with x.
\[du = \dfrac{d}{{dx}}\dfrac{{\cos x}}{{\sin x}}\]
With the help of product and quotient rule,
\[{\left( {\dfrac{u}{v}} \right)^1} = \dfrac{{v.{u^1} - u.{v^1}}}{{{v^2}}}\]
\[du = \dfrac{{ - \sin x.\sin x - \cos x.\cos x}}{{{{\sin }^2}x}}dx\]
Taking the product of the terms,
\[du = \dfrac{{ - {{\sin }^2}x - {{\cos }^2}x}}{{{{\sin }^2}x}}dx\]
Taking minus common from numerator terms,
\[du = \dfrac{{ - \left( {{{\sin }^2}x + {{\cos }^2}x} \right)}}{{{{\sin }^2}x}}dx\]
We know the identity \[{\sin ^2}x + {\cos ^2}x = 1\] then
\[du = \dfrac{{ - 1}}{{{{\sin }^2}x}}dx\]
\[ - du = \dfrac{1}{{{{\sin }^2}x}}dx\]
Now substituting the integral,
\[ = \int { - du} \]
Taking minus outside
\[ = - \int {du} \]
Taking integral,
\[ = - u\]
Substituting value of u,
\[ = - cotx\]
This is the answer.
So, the correct answer is “- cotx”.
Note: Note that this approach is used because no other approach is as easy as this. Also not that cot function is purposely used because only that function has the function and derivative are in numerator and denominator form. That helps to lead the problem in a smoother way. We can go for the ways like
\[\dfrac{1}{{{{\sin }^2}x}} = \dfrac{1}{{1 - {{\cos }^2}x}}dx\]
Or any other identity rearrangement but that will only lengthen the problem or can never lead to the answer. So do prefer this solution.
Complete step by step solution:
Given that find the integral of \[\dfrac{1}{{{{\sin }^2}x}}\]
So that is \[\int {\dfrac{1}{{{{\sin }^2}x}}dx} \]
Now we will substitute \[u = \cot x\]
On writing \[\cot x = \dfrac{{\cos x}}{{\sin x}}\]
Next we will differentiate the above terms with x.
\[du = \dfrac{d}{{dx}}\dfrac{{\cos x}}{{\sin x}}\]
With the help of product and quotient rule,
\[{\left( {\dfrac{u}{v}} \right)^1} = \dfrac{{v.{u^1} - u.{v^1}}}{{{v^2}}}\]
\[du = \dfrac{{ - \sin x.\sin x - \cos x.\cos x}}{{{{\sin }^2}x}}dx\]
Taking the product of the terms,
\[du = \dfrac{{ - {{\sin }^2}x - {{\cos }^2}x}}{{{{\sin }^2}x}}dx\]
Taking minus common from numerator terms,
\[du = \dfrac{{ - \left( {{{\sin }^2}x + {{\cos }^2}x} \right)}}{{{{\sin }^2}x}}dx\]
We know the identity \[{\sin ^2}x + {\cos ^2}x = 1\] then
\[du = \dfrac{{ - 1}}{{{{\sin }^2}x}}dx\]
\[ - du = \dfrac{1}{{{{\sin }^2}x}}dx\]
Now substituting the integral,
\[ = \int { - du} \]
Taking minus outside
\[ = - \int {du} \]
Taking integral,
\[ = - u\]
Substituting value of u,
\[ = - cotx\]
This is the answer.
So, the correct answer is “- cotx”.
Note: Note that this approach is used because no other approach is as easy as this. Also not that cot function is purposely used because only that function has the function and derivative are in numerator and denominator form. That helps to lead the problem in a smoother way. We can go for the ways like
\[\dfrac{1}{{{{\sin }^2}x}} = \dfrac{1}{{1 - {{\cos }^2}x}}dx\]
Or any other identity rearrangement but that will only lengthen the problem or can never lead to the answer. So do prefer this solution.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)