Mathematics

Evaluate $$\int { \sin ^{ 3 }{ x }  } \sin { 2x } \, dx$$.


SOLUTION
$$\displaystyle\int{{\sin}^{3}{x}\sin{2x}dx}$$
$$=\displaystyle\int{{\sin}^{3}{x}2\sin{x}\cos{x}dx}$$
$$I=2\displaystyle\int{{\sin}^{4}{x}\cos{x}dx}$$

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

$$I=2\displaystyle\int{{t}^{4}dt}$$

$$=2\left(\dfrac{{t}^{5}}{5}\right)+c$$ , where $$c$$ is the constant of integration

$$=\dfrac{2 \, {\sin}^{5}{x}}{5}+c$$ , where $$t=\sin{x}$$
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Subjective Medium Published on 17th 09, 2020
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