\(\displaystyle \int 8 \sin^{2}(x) \cos^{3}(x) dx\)
\(\displaystyle \int 8\sin^{2}{x}(\cos^{2}(x))(\cos(x))\)
\(\displaystyle \int 8\sin^{2}{x}(\cos^{2}(x))(\cos(x)\)
\(\displaystyle \int 8\sin^{2}{x}(1 - \sin^{2}(x))(\cos(x)\)
\(\displaystyle u = \sin(x)\)
\(\displaystyle du = \cos(x) dx\)
\(\displaystyle \int (8u^{2})(1 - u^{2})]du\)
\(\displaystyle \int (8u^{2} - 8u^{4})]du\)
\(\displaystyle = 8\dfrac{u^{3}}{3} - 8\dfrac{u^{5}}{5} + C\)
\(\displaystyle = 8\dfrac{\sin^{3}(x)}{3} - 8\dfrac{\sin^{5}(x)}{5} + C\)
\(\displaystyle \int 8\sin^{2}{x}(\cos^{2}(x))(\cos(x))\)
\(\displaystyle \int 8\sin^{2}{x}(\cos^{2}(x))(\cos(x)\)
\(\displaystyle \int 8\sin^{2}{x}(1 - \sin^{2}(x))(\cos(x)\)
\(\displaystyle u = \sin(x)\)
\(\displaystyle du = \cos(x) dx\)
\(\displaystyle \int (8u^{2})(1 - u^{2})]du\)
\(\displaystyle \int (8u^{2} - 8u^{4})]du\)
\(\displaystyle = 8\dfrac{u^{3}}{3} - 8\dfrac{u^{5}}{5} + C\)
\(\displaystyle = 8\dfrac{\sin^{3}(x)}{3} - 8\dfrac{\sin^{5}(x)}{5} + C\)
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