Mathematics

$$\int  sin^{2/3}x cos^{3}x dx$$


SOLUTION
$$\displaystyle\int \sin^{2/3}x\cos^3xdx$$
$$=\displaystyle\int \sin^{2/3}x\cos^2x\cos xdx$$
$$=\displaystyle\int \sin^{2/3}x(1-\sin^2x)\cos xdx$$
$$=\displaystyle\int \sin^{2/3}x\cos xdx-\displaystyle\int \sin^{2/3+2}x\cos xdx$$
$$=\displaystyle\int \sin^{2/3}x\cos xdx-\displaystyle\int \sin^{5/2}x\cos xdx$$
Let $$t=\sin x$$
$$\Rightarrow dt=\cos xdx$$
$$=\displaystyle\int t^{2/3}dt-\displaystyle\int t^{5/2}dt$$
$$=\dfrac{t^{2/3+1}}{2/3+1}-\dfrac{t^{5/2+1}}{5/2+1}+c$$
$$=\dfrac{t^{5/2}}{5/3}-\dfrac{t^{7/2}}{7/2}+c$$ where c is the constant of integration
$$=\dfrac{3(\sin x)^{5/3}}{5}-\dfrac{2(\sin x)^{7/2}}{7}+c$$
$$=\dfrac{3\sin^{5/3}x}{5}-\dfrac{2}{7}\sin^{7/2}x+c$$ where $$t=\sin x$$.
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Subjective Medium Published on 17th 09, 2020
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