Integrate the function ^3 x e^ x - Vidyadip

Question

Integrate the function $\int {{{\cos }^3}x{e^{\log \sin x}}}$

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Answer

$I=\int {{{\cos }^3}x.{e^{\log \sin x}}} dx$

$\because {e^{\log \theta }} = \theta $

$\therefore {e^{\log \sin x}} = \sin x$

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

Put cos x = t

$ - \sin x\,dx = dt$

$\sin x\,dx = - dt$

$I = \int { - {t^3}dt} $

$ = - \frac{{{t^4}}}{4} + c = - \frac{{{{\cos }^4}x}}{4} + c$

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