Mathematics

Evaluate the following integral:
$$\int { \cfrac { \sin ^{ 3 }{ x }  }{ \sqrt { \cos { x }  }  }  } dx\quad $$


SOLUTION
$$\displaystyle\int{\dfrac{{\sin}^{3}{x}}{\sqrt{\cos{x}}}dx}$$

$$=\displaystyle\int{\dfrac{\left(1-{\cos}^{2}{x}\right)\sin{x}}{\sqrt{\cos{x}}}dx}$$

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

$$=-\displaystyle\int{\dfrac{\left(1-{t}^{2}\right)dt}{\sqrt{t}}}$$

$$=-\displaystyle\int{\dfrac{dt}{\sqrt{t}}}+\displaystyle\int{\dfrac{{t}^{2}\,dt}{\sqrt{t}}}$$

$$=-\displaystyle\int{{t}^{-\frac{1}{2}}dt}+\displaystyle\int{{t}^{2-\frac{1}{2}}\,dt}$$

$$=-\displaystyle\int{{t}^{-\frac{1}{2}}dt}+\displaystyle\int{{t}^{\frac{3}{2}}\,dt}$$

$$=-\dfrac{{t}^{-\frac{1}{2}+1}}{-\dfrac{1}{2}+1}+\dfrac{{t}^{\frac{3}{2}+1}}{\dfrac{3}{2}+1}+c$$

$$=-\dfrac{{t}^{\frac{1}{2}}}{\dfrac{1}{2}}+\dfrac{{t}^{\frac{5}{2}}}{\dfrac{5}{2}}+c$$

$$=-{2}{t}^{\frac{1}{2}}+\dfrac{2}{5}{t}^{\frac{5}{2}}+c$$

$$=-{2}{\cos}^{\frac{1}{2}}{x}+\dfrac{2}{5}{cos}^{\frac{5}{2}}{x}+c$$ where 

$$t=\cos{x}$$

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Subjective Medium Published on 17th 09, 2020
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