Integral Calculus: Integral of csc^3 cos^3 x dx

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Integral Calculus: Integral of csc^3 cos^3 x dx

Integral of csc^3 cos^3 xdx

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$\displaystyle \int \csc^3 x

$\displaystyle \int \csc^3 x \cos^3 x \, dx$

      $= \displaystyle \int \dfrac{1}{\sin^3 x} \cdot \cos^2 x \cdot \cos x \, dx$

      $= \displaystyle \int (\sin x)^{-3} \cdot (1 - \sin^2 x) \cdot \cos x \, dx$

      $= \displaystyle \int (\sin x)^{-3} (\cos x \, dx) - \int (\sin x)^{-1}(\cos x \, dx)$

      $= \displaystyle \int (\sin x)^{-3} (\cos x \, dx) - \int \dfrac{\cos x \, dx}{\sin x}$

      $= \displaystyle \int (\sin x)^{-3} (\cos x \, dx) - \int \cot x \, dx$
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