Mathematics

Evaluate :
$$\int { { sin }^{ 5 }x.dx } $$


SOLUTION
$$\displaystyle\int{{\sin}^{5}{x}dx}$$
$$=\displaystyle\int{{\sin}^{4}{x}\sin{x}dx}$$
$$=\displaystyle\int{{\left(1-{\cos}^{2}{x}\right)}^{2}\sin{x}dx}$$
Let $$u=\cos{x}\Rightarrow du=-\sin{x}dx$$
$$=-\displaystyle\int{{\left(1-{u}^{2}\right)}^{2}du}$$
$$=-\displaystyle\int{\left(1+{u}^{4}-2{u}^{2}\right)du}$$
$$=-\left[u+\dfrac{{u}^{5}}{5}-\dfrac{2{u}^{3}}{3}\right]+c$$
$$=-\cos{x}-\dfrac{{\cos}^{5}{x}}{5}+\dfrac{2{\cos}^{3}{x}}{3}-c$$
$$=-\dfrac{{\cos}^{5}{x}}{5}+\dfrac{2{\cos}^{3}{x}}{3}-\cos{x}-c$$
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Subjective Medium Published on 17th 09, 2020
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