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
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