Answer
350.1k+ views
Hint: We are asked to find the integral of the given term. Since this is a complex one we need to first solve the stuff inside the bracket and we need to simplify that. In this case we will use several identities of double angle involving cosine and sine and then after simplifying the matter inside the bracket we will integrate the whole term. We should be aware of all the trigonometric identities.
Complete step-by-step solution:
We have $\int{{{\tan }^{-1}}\left( \dfrac{\sin 2x}{1+\cos 2x} \right)dx}$.
The term inside the bracket is:
$\dfrac{\sin 2x}{1+\cos 2x}$
We can use the following identity in place of $sin2x$:
$\sin 2x=2\sin x\cos x$
And in place of $cos2x$, we can use the following formula:
$\cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x$
Putting in the fraction and simplifying it, we have:
$\dfrac{\sin 2x}{1+\cos 2x}=\dfrac{2\sin x\cos x}{1+\left( {{\cos }^{2}}x-{{\sin }^{2}}x \right)}$
Now, we use the following important identity:
${{\cos }^{2}}x+{{\sin }^{2}}x=1$
Using the same we get:
${{\cos }^{2}}x=1-{{\sin }^{2}}x$
Putting this in the denominator:
$\dfrac{\sin 2x}{1+\cos 2x}=\dfrac{2\sin x\cos x}{2{{\cos }^{2}}x}$
So, we can say that:
$\dfrac{\sin 2x}{1+\cos 2x}=\tan x$
We obtained this using the formula:
$\dfrac{\sin x}{\cos x}=\tan x$
Putting this term inside the bracket we have:
$\int{{{\tan }^{-1}}\left( \tan x \right)dx=\int{xdx}}$
And we know that:
\[\int{xdx=\dfrac{{{x}^{2}}}{2}+c}\]
Hence, option $\left( b \right)\dfrac{{{x}^{2}}}{2}+c$ is correct.
Note: If the formulae related to the double angle of sine and cosine are not known then this becomes a very difficult integral to operate because the terms inside bracket cannot be solved so easily. Moreover, you should be aware about the integrals of some basic algebraic terms such as the polynomial terms.
Complete step-by-step solution:
We have $\int{{{\tan }^{-1}}\left( \dfrac{\sin 2x}{1+\cos 2x} \right)dx}$.
The term inside the bracket is:
$\dfrac{\sin 2x}{1+\cos 2x}$
We can use the following identity in place of $sin2x$:
$\sin 2x=2\sin x\cos x$
And in place of $cos2x$, we can use the following formula:
$\cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x$
Putting in the fraction and simplifying it, we have:
$\dfrac{\sin 2x}{1+\cos 2x}=\dfrac{2\sin x\cos x}{1+\left( {{\cos }^{2}}x-{{\sin }^{2}}x \right)}$
Now, we use the following important identity:
${{\cos }^{2}}x+{{\sin }^{2}}x=1$
Using the same we get:
${{\cos }^{2}}x=1-{{\sin }^{2}}x$
Putting this in the denominator:
$\dfrac{\sin 2x}{1+\cos 2x}=\dfrac{2\sin x\cos x}{2{{\cos }^{2}}x}$
So, we can say that:
$\dfrac{\sin 2x}{1+\cos 2x}=\tan x$
We obtained this using the formula:
$\dfrac{\sin x}{\cos x}=\tan x$
Putting this term inside the bracket we have:
$\int{{{\tan }^{-1}}\left( \tan x \right)dx=\int{xdx}}$
And we know that:
\[\int{xdx=\dfrac{{{x}^{2}}}{2}+c}\]
Hence, option $\left( b \right)\dfrac{{{x}^{2}}}{2}+c$ is correct.
Note: If the formulae related to the double angle of sine and cosine are not known then this becomes a very difficult integral to operate because the terms inside bracket cannot be solved so easily. Moreover, you should be aware about the integrals of some basic algebraic terms such as the polynomial terms.
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)