Mathematics

Solve the integral:-
$$\int {{{\cos }^3}x\, {{\sin }^5}xdx = ?} $$


SOLUTION
$$\displaystyle\int{{\cos}^{3}{x}{\sin}^{5}{x}dx}$$

$$=\displaystyle\int{\cos{x}{\cos}^{2}{x}{\sin}^{5}{x}dx}$$

$$=\displaystyle\int{\cos{x}\left(1-{\sin}^{2}{x}\right){\sin}^{5}{x}dx}$$

$$=\displaystyle\int{\left({\sin}^{5}{x}-{\sin}^{7}{x}\right)\cos{x}dx}$$

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

$$=\displaystyle\int{\left({t}^{5}-{t}^{7}\right)dt}$$

$$=\dfrac{{t}^{6}}{6}-\dfrac{{t}^{8}}{8}+c$$

$$=\dfrac{{\sin}^{6}{x}}{6}-\dfrac{{\sin}^{8}{x}}{8}+c$$

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Subjective Medium Published on 17th 09, 2020
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