\(\displaystyle \int_{0}^{\pi/2} \sin^{11}\theta \cos^{5}\theta d\theta \)
\(\displaystyle \int_{0}^{\pi/2} \sin^{11}\theta (\cos^{2}\theta)^{2} (\cos \theta) d\theta \)
\(\displaystyle \int_{0}^{\pi/2} \sin^{11}\theta (\sin^{2}\theta - 1)^{2} (\cos \theta) d\theta \)
\(\displaystyle u = \sin \theta\)
\(\displaystyle uu = \cos \theta d\theta\)
\(\displaystyle \int_{0}^{\pi/2} u^{11} (u^{2}- 1)^{2} du \)
\(\displaystyle \int_{0}^{\pi/2} u^{11} (u^{4} - 2u^{2} + 1) du \)
\(\displaystyle \int_{0}^{\pi/2} u^{14} - 2u^{13} + u^{11 } du \)
\(\displaystyle = \dfrac{u^{15}}{15} - \dfrac{2u^{14}}{14} + \dfrac{u^{12}}{12}\) evaluated at \(\displaystyle [0,\dfrac{\pi}{2}]\)
\(\displaystyle = \dfrac{\sin^{15}(\theta)}{15} - \dfrac{2\sin^{14}(\theta)}{14} + \dfrac{\sin^{12}(\theta)}{12}\) evaluated at \(\displaystyle [0,\dfrac{\pi}{2}]\)
\(\displaystyle = [\dfrac{\sin^{15}(\pi/2)}{15} - \dfrac{2\sin^{14}(\pi/2)}{14} + \dfrac{\sin^{12}(\pi/2)}{12}] - [\dfrac{\sin^{15}(0)}{15} - \dfrac{2\sin^{14}(0)}{14} + \dfrac{\sin^{12}(0)}{12}] = \dfrac{1}{140} \)
\(\displaystyle \int_{0}^{\pi/2} \sin^{11}\theta (\cos^{2}\theta)^{2} (\cos \theta) d\theta \)
\(\displaystyle \int_{0}^{\pi/2} \sin^{11}\theta (\sin^{2}\theta - 1)^{2} (\cos \theta) d\theta \)
\(\displaystyle u = \sin \theta\)
\(\displaystyle uu = \cos \theta d\theta\)
\(\displaystyle \int_{0}^{\pi/2} u^{11} (u^{2}- 1)^{2} du \)
\(\displaystyle \int_{0}^{\pi/2} u^{11} (u^{4} - 2u^{2} + 1) du \)
\(\displaystyle \int_{0}^{\pi/2} u^{14} - 2u^{13} + u^{11 } du \)
\(\displaystyle = \dfrac{u^{15}}{15} - \dfrac{2u^{14}}{14} + \dfrac{u^{12}}{12}\) evaluated at \(\displaystyle [0,\dfrac{\pi}{2}]\)
\(\displaystyle = \dfrac{\sin^{15}(\theta)}{15} - \dfrac{2\sin^{14}(\theta)}{14} + \dfrac{\sin^{12}(\theta)}{12}\) evaluated at \(\displaystyle [0,\dfrac{\pi}{2}]\)
\(\displaystyle = [\dfrac{\sin^{15}(\pi/2)}{15} - \dfrac{2\sin^{14}(\pi/2)}{14} + \dfrac{\sin^{12}(\pi/2)}{12}] - [\dfrac{\sin^{15}(0)}{15} - \dfrac{2\sin^{14}(0)}{14} + \dfrac{\sin^{12}(0)}{12}] = \dfrac{1}{140} \)
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