int sin ^3(x) cdot cos ^3(x) , dx = | vedclass

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(C) We need to evaluate the integral $I = \int \sin ^3(x) \cdot \cos ^3(x) \, dx$. Rewrite the integral as: $I = \int \sin ^3(x) \cdot \cos ^2(x) \cdo

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$\frac{1}{4} \sin ^4(x) - \frac{1}{6} \sin ^6(x) + C$

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(C) We need to evaluate the integral $I = \int \sin ^3(x) \cdot \cos ^3(x) \, dx$. Rewrite the integral as: $I = \int \sin ^3(x) \cdot \cos ^2(x) \cdot \cos(x) \, dx$ Using the identity $\cos ^2(x) = 1 - \sin ^2(x)$,we get: $I = \int \sin ^3(x) \cdot (1 - \sin ^2(x)) \cdot \cos(x) \, dx$ Let $t = \sin(x)$,then $dt = \cos(x) \, dx$. Substituting these into the integral: $I = \int t^3(1 - t^2) \, dt = \int (t^3 - t^5) \, dt$ Integrating with respect to $t$: $I = \frac{t^4}{4} - \frac{t^6}{6} + C$ Substituting back $t = \sin(x)$: $I = \frac{1}{4} \sin ^4(x) - \frac{1}{6} \sin ^6(x) + C$

$\int \sin ^3(x) \cdot \cos ^3(x) \, dx =$

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