7.1 Integration By Parts/45
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \int_{0}^{\frac{\pi}{2}} sin^n(x) dx = \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} sin^{n-2} (x) dx \\[2ex] &= -\frac{1}{n} cos(x) sin^{n-1} (x) + \frac{n-1}{n} \bigg|_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} sin^{n-2}(x) dx \\[2ex] &= -\frac{1}{n} cos(\frac{\pi}{2}) sin^{n-1} (\frac{\pi}{2})+ \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} sin^{n-2}(x) dx \\[2ex] &= -\frac{1}{n} (0) (1) + \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} sin^{n-2}(x) dx \\[2ex] &= \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} sin^{n-2}(x) dx \\[2ex] \end{align} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \int_{0}^{\frac{\pi}{2}} sin^3(x) dx \\[2ex] &= \frac{3-1}{3} \int_{0}^{\frac{\pi}{2}} sin(x) dx \\[2ex] &= \frac{2}{3} \int_{0}^{\frac{\pi}{2}} sin(x) dx \\[2ex] &= \frac{2}{3} [-cos(x) \bigg|_{0}^{\frac{\pi}{2}} \\[2ex] &= \frac{2}{3} [-cos(\frac{\pi}{2}) -(-cos(0))] \\[2ex] &= \frac{2}{3} [-1(0) -(-1)] \\[2ex] &= \frac{2}{3}[1] \\[2ex] &= \frac{2}{3} \\[2ex] \end{align} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \int_{0}^{\frac{\pi}{2}} sin^{2n+1} (x) dx \\[2ex] &= \frac{2n+1-1}{2n+1} \int_{0}^{\frac{\pi}{2}} sin^{2n+1-2} (x) dx \\[2ex] &= \frac{2n}{2n+1} \int_{0}^{\frac{\pi}{2}} sin^{2n-1} (x) dx \\[2ex] \end{align} }
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{0}^{\frac {\pi }{2}}sin^{5}(x)dx&={\frac {5-1}{5}}\int _{0}^{\frac {\pi }{2}}sin^{3}(x)dx\\[2ex]&={\frac {4}{5}}\int _{0}^{\frac {\pi }{2}}(1-cos^{2}(x))sin(x)dx\\[2ex]&=-\int _{1}^{0}(1-u^{2})\cdot du\\[2ex]&={\frac {4}{5}}\left[u-{\frac {u^{3}}{3}}\right]{\bigg |}_{0}^{1}\\[2ex]&u=cos(x)\\[2ex]&du=-sin(x)\\[2ex]&dx={\frac {du}{-sin(x)}}\\[2ex]&\int sin^{3}(x)dx\\[2ex]&={\frac {4}{5}}[1-{\frac {1}{3}}]\\[2ex]&={\frac {8}{15}}\end{aligned}}}