Mathematics

Solve:

$$\int {{{\sin }^3}x.{{\cos }^2}xdx} $$.


SOLUTION
$$\displaystyle\int \sin^3x(\cos^2x)dx$$

$$\displaystyle\int \sin x(\sin^2x)(\cos^2x)dx$$
$$\Rightarrow \displaystyle\int \sin x(1-\cos^2x)(\cos^2x)dx$$

$$\Rightarrow \displaystyle\int \sin x\cos^2xdx-\displaystyle\int \sin x\cos^4xdx$$

Put $$\cos x=t$$
$$-\sin xdx=dt$$
$$\Rightarrow -\displaystyle\int t^2dt+\displaystyle\int t^4dt$$
$$\Rightarrow -\dfrac{t^3}{3}+\dfrac{t^5}{5}+c$$
$$\displaystyle\int \sin^3x\cos xdx=\dfrac{\cos^5x}{5}-\dfrac{\cos^3x}{3}+c$$.
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Subjective Medium Published on 17th 09, 2020
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