Question

find the integration of sin^2/3 x cos^3 x​

Answer 1

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$\int [sin(x)]^{\frac{2}{3}} cos^3(x)dx$

$= \frac{3}{5}{\left[sin(x)\right]}^{\frac{5}{3}} - \frac{3}{11}{\left[sin(x)\right]}^{\frac{11}{3}} +C$

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$I = \int [sin(x)]^{\frac{2}{3}} cos^3(x)dx$

$I = \int [sin(x)]^{\frac{2}{3}} cos^2(x)cos(x)dx$

$I = \int [sin(x)]^{\frac{2}{3}}[1-sin^2(x)]cos(x)dx$

Substitute $sin(x) = t$

$\to cos(x)dx = dt$

$I = \int {t} ^{\frac{2}{3}}[1-t^2]dt$

$I = \int [{t}^{\frac{2}{3}} -{t}^{\frac{8}{3}}]dt$

$I = \left[\frac{{t}^{\frac{2}{3}+1}}{\frac{2}{3}+1} -\frac{{t}^{\frac{8}{3}+1}}{\frac{8}{3}+1}\right]+C$

$I = \left[\frac{{t}^{\frac{5}{3}}}{\frac{5}{3}} -\frac{{t}^{\frac{11}{3}}}{\frac{11}{3}}\right]+C$

$I = \left[\frac{3}{5}{t}^{\frac{5}{3}} - \frac{3}{11}{t}^{\frac{11}{3}}\right]+C$

$I = \frac{3}{5}{\left[sin(x)\right]}^{\frac{5}{3}} - \frac{3}{11}{\left[sin(x)\right]}^{\frac{11}{3}}+C$