Answer
Verified
414.9k+ views
Hint: We will add and subtract 1 in numerator as $\int{\dfrac{\cos x+1-1}{1+\cos x}dx}$ and then separate it as $\int{\dfrac{1+\cos x}{1+\cos x}dx-\int{\dfrac{1}{1+\cos x}dx}}$ and then solve accordingly. We will also use few trigonometric formula such as $\cos 2\theta =2{{\cos }^{2}}\theta -1$ and $\cos 2\theta +1=2{{\cos }^{2}}\theta $.
Complete step-by-step answer:
We have given that to integrate $\int{\dfrac{\cos x}{1+\cos x}dx}$. We will add and subtract 1 in the numerator we get $\int{\dfrac{\cos x+1-1}{1+\cos x}dx}$. Now we split the integration into two simplified integration and solve them, independently, $\int{\dfrac{1+\cos x}{1+\cos x}dx-\int{\dfrac{1}{1+\cos x}dx}}$.
On further solving we get $\int{1dx-\int{\dfrac{1}{1+\cos x}dx}}$. Now we know that $\cos 2\theta =2{{\cos }^{2}}\theta -1$ and $\cos 2\theta +1=2{{\cos }^{2}}\theta $ on replacing $\theta $ with $\dfrac{x}{2}$, we get $\cos 2\dfrac{x}{2}=2{{\cos }^{2}}\dfrac{x}{2}-1$, further simplifying $\operatorname{cosx}+1=2{{\cos }^{2}}\dfrac{x}{2}$. So, on putting $1+\cos x=2{{\cos }^{2}}\dfrac{x}{2}$, we get $\int{\left( 1 \right)dx-\int{\dfrac{1}{2{{\cos }^{2}}\dfrac{x}{2}}dx}}$.
We know that $\cos \theta =\dfrac{1}{\sec x}$, thus ${{\cos }^{2}}\dfrac{x}{2}$ can be written as $\dfrac{1}{{{\sec }^{2}}\dfrac{x}{2}}$ , we get $\int{\left( 1 \right)dx-\dfrac{1}{2}\int{{{\sec }^{2}}\dfrac{x}{2}dx}}$.
We know that $\int{{{\sec }^{2}}\left( ax+b \right)dx=\dfrac{\tan \left( ax+b \right)}{a}}+c$, we get = $x-\dfrac{1}{2}\dfrac{\tan \left( \dfrac{x}{2} \right)}{\dfrac{1}{2}}+c$ simplifying further, we get our final answer as = $x-\tan \dfrac{x}{2}+c$.
Note: Usually students make mistakes in the last step in the integration of $\int{{{\sec }^{2}}\dfrac{x}{2}}$. Most of the student directly integrate $\int{{{\sec }^{2}}\dfrac{x}{2}}$ as $\tan \dfrac{x}{2}+c$, which is not correct. The correct integration of $\int{{{\sec }^{2}}\dfrac{x}{2}}$ is$\dfrac{\tan \dfrac{x}{2}}{\dfrac{1}{2}}+c$. Also, student may forget the sub trigonometric formulas like $\cos 2\theta =2{{\cos }^{2}}\theta -1$ thus, it is recommended to memorize all the formulas of trigonometry before solving such questions.
Complete step-by-step answer:
We have given that to integrate $\int{\dfrac{\cos x}{1+\cos x}dx}$. We will add and subtract 1 in the numerator we get $\int{\dfrac{\cos x+1-1}{1+\cos x}dx}$. Now we split the integration into two simplified integration and solve them, independently, $\int{\dfrac{1+\cos x}{1+\cos x}dx-\int{\dfrac{1}{1+\cos x}dx}}$.
On further solving we get $\int{1dx-\int{\dfrac{1}{1+\cos x}dx}}$. Now we know that $\cos 2\theta =2{{\cos }^{2}}\theta -1$ and $\cos 2\theta +1=2{{\cos }^{2}}\theta $ on replacing $\theta $ with $\dfrac{x}{2}$, we get $\cos 2\dfrac{x}{2}=2{{\cos }^{2}}\dfrac{x}{2}-1$, further simplifying $\operatorname{cosx}+1=2{{\cos }^{2}}\dfrac{x}{2}$. So, on putting $1+\cos x=2{{\cos }^{2}}\dfrac{x}{2}$, we get $\int{\left( 1 \right)dx-\int{\dfrac{1}{2{{\cos }^{2}}\dfrac{x}{2}}dx}}$.
We know that $\cos \theta =\dfrac{1}{\sec x}$, thus ${{\cos }^{2}}\dfrac{x}{2}$ can be written as $\dfrac{1}{{{\sec }^{2}}\dfrac{x}{2}}$ , we get $\int{\left( 1 \right)dx-\dfrac{1}{2}\int{{{\sec }^{2}}\dfrac{x}{2}dx}}$.
We know that $\int{{{\sec }^{2}}\left( ax+b \right)dx=\dfrac{\tan \left( ax+b \right)}{a}}+c$, we get = $x-\dfrac{1}{2}\dfrac{\tan \left( \dfrac{x}{2} \right)}{\dfrac{1}{2}}+c$ simplifying further, we get our final answer as = $x-\tan \dfrac{x}{2}+c$.
Note: Usually students make mistakes in the last step in the integration of $\int{{{\sec }^{2}}\dfrac{x}{2}}$. Most of the student directly integrate $\int{{{\sec }^{2}}\dfrac{x}{2}}$ as $\tan \dfrac{x}{2}+c$, which is not correct. The correct integration of $\int{{{\sec }^{2}}\dfrac{x}{2}}$ is$\dfrac{\tan \dfrac{x}{2}}{\dfrac{1}{2}}+c$. Also, student may forget the sub trigonometric formulas like $\cos 2\theta =2{{\cos }^{2}}\theta -1$ thus, it is recommended to memorize all the formulas of trigonometry before solving such questions.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE