Mathematics

# $\int { \cos ^{ 2 }{ \cfrac { x }{ 2 } } } dx$

##### SOLUTION
$\displaystyle\int{{\cos}^{2}{\dfrac{x}{2}}\,dx}$

$=\dfrac{1}{2}\displaystyle\int{2{\cos}^{2}{\dfrac{x}{2}}\,dx}$
We know that $\cos{x}=2{\cos}^{2}{\dfrac{x}{2}}-1\Rightarrow\,2{\cos}^{2}{\dfrac{x}{2}}=1+\cos{x}$

$\therefore\,\dfrac{1}{2}\displaystyle\int{{\cos}^{2}{\dfrac{x}{2}}\,dx}$

$=\dfrac{1}{2}\displaystyle\int{\left(1+\cos{x}\right)dx}$

Let $t=\sin{x}\Rightarrow\,dt=\cos{x}dx$

$=\dfrac{1}{2}\displaystyle\int{dx}+\dfrac{1}{2}\displaystyle\int{\cos{x}dx}$

$=\dfrac{1}{2}\left(x+\sin{x}\right)+c$ where $c$ is the constant of integration.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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