Answer
64.8k+ views
Hint: For this question, first open the bracket, then separate it into two parts. Suppose individual parts in the form of variables like m, n and at last apply integration formulae.
Complete step by step solution:
The given integrand is
$\int{\left( 1-\cos x \right)\cos e{{c}^{2}}}x dx$ bracket with
First we will open the brackets with help of multiplication then we have to divide it into two terms or parts and integrate individual parts , so that
$\int{\left( 1-\cos x \right)\cos e{{c}^{2}}}x dx=$$\int{\cos e{{c}^{2}}}x dx-\int{\cos x.\cos e{{c}^{2}}}x dx$……………………..(i)
Now to integrate it, let us suppose $\sin x= m$
If we differentiate $\sin x=m$, we get by the differentiation formula,
$\begin{align}
& \dfrac{d}{dx}\sin x=dm \\
& \rightarrow \cos x dx=dm \\
\end{align}$
We know that $\cos e{{c}^{2}}x dx=-\cot x$
On putting these values in equation (i)
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$-\cot x-\int{\dfrac{dm}{{{m}^{2}}}}$
By integration formula, we know that $\int{{{x}^{n}}}dx=\dfrac{{{x}^{n+1}}}{n+1}+c$, where c is constant of integration.
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$ =$-\cot x+\dfrac{1}{m}$
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$ =$-\cot x+\dfrac{1}{\sin x}$
Because we already supposed that $\sin x=m$
Again we can write $\cot x=\dfrac{\cos x}{\sin x}$, then
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$-\dfrac{\cos x}{\sin x}+\dfrac{1}{\sin x}$
By common denominator rule of subtraction,
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$-\dfrac{\cos x+1}{\sin x}=\dfrac{1-\cos x}{\sin x}$
By the identity of trigonometry,
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$\dfrac{2{{\sin }^{2}}\dfrac{x}{2}}{2\sin \dfrac{x}{2}\cos \dfrac{x}{2}}$
$\sin x/2\,\,cancel\,out\,to\,\sin x/2$
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$\dfrac{\sin \dfrac{x}{2}}{\cos \dfrac{x}{2}}$
We know that$\dfrac{\sin \dfrac{x}{2}}{\cos \dfrac{x}{2}}=\tan \dfrac{x}{2}$, then we get
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$\tan \dfrac{x}{2}+c$
Where c is constant of integration.
Hence, The option (B) is the correct option.
Additional information: If more than one constant of integration is used while solving the integral, then at the end of the solution write only one constant of integration.
If the denominators of two fractions when they are in addition or subtraction, then we don’t need to take LCM .
Note: Sometimes students make mistakes, they apply the integration formula instead of before opening the brackets, then their solution would be wrong. It's important to solve such types of questions with the help of formulae and identities.
Complete step by step solution:
The given integrand is
$\int{\left( 1-\cos x \right)\cos e{{c}^{2}}}x dx$ bracket with
First we will open the brackets with help of multiplication then we have to divide it into two terms or parts and integrate individual parts , so that
$\int{\left( 1-\cos x \right)\cos e{{c}^{2}}}x dx=$$\int{\cos e{{c}^{2}}}x dx-\int{\cos x.\cos e{{c}^{2}}}x dx$……………………..(i)
Now to integrate it, let us suppose $\sin x= m$
If we differentiate $\sin x=m$, we get by the differentiation formula,
$\begin{align}
& \dfrac{d}{dx}\sin x=dm \\
& \rightarrow \cos x dx=dm \\
\end{align}$
We know that $\cos e{{c}^{2}}x dx=-\cot x$
On putting these values in equation (i)
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$-\cot x-\int{\dfrac{dm}{{{m}^{2}}}}$
By integration formula, we know that $\int{{{x}^{n}}}dx=\dfrac{{{x}^{n+1}}}{n+1}+c$, where c is constant of integration.
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$ =$-\cot x+\dfrac{1}{m}$
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$ =$-\cot x+\dfrac{1}{\sin x}$
Because we already supposed that $\sin x=m$
Again we can write $\cot x=\dfrac{\cos x}{\sin x}$, then
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$-\dfrac{\cos x}{\sin x}+\dfrac{1}{\sin x}$
By common denominator rule of subtraction,
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$-\dfrac{\cos x+1}{\sin x}=\dfrac{1-\cos x}{\sin x}$
By the identity of trigonometry,
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$\dfrac{2{{\sin }^{2}}\dfrac{x}{2}}{2\sin \dfrac{x}{2}\cos \dfrac{x}{2}}$
$\sin x/2\,\,cancel\,out\,to\,\sin x/2$
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$\dfrac{\sin \dfrac{x}{2}}{\cos \dfrac{x}{2}}$
We know that$\dfrac{\sin \dfrac{x}{2}}{\cos \dfrac{x}{2}}=\tan \dfrac{x}{2}$, then we get
$\int{\cos e{{c}^{2}}}xdx-\int{\cos x.\cos e{{c}^{2}}}xdx$=$\tan \dfrac{x}{2}+c$
Where c is constant of integration.
Hence, The option (B) is the correct option.
Additional information: If more than one constant of integration is used while solving the integral, then at the end of the solution write only one constant of integration.
If the denominators of two fractions when they are in addition or subtraction, then we don’t need to take LCM .
Note: Sometimes students make mistakes, they apply the integration formula instead of before opening the brackets, then their solution would be wrong. It's important to solve such types of questions with the help of formulae and identities.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)