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} }

${\begin{aligned}\int _{0}^{\frac {\pi }{2}}sin^{5}(x)dx\\[2ex]&={\frac {5-1}{5}}\int _{0}^{\frac {\pi }{2}}sin^{3}(x)dx\\[2ex]&u=cos(x)\\[2ex]&du=-sin(x)\\[2ex]&dx={\frac {du}{-sin(x)}}\\[2ex]&\int sin^{3}(x)dx\\[2ex]&=\int (1-cos^{2}(x))sin(x)dx\\[2ex]&=\int (1-u^{2})sin(x)\cdot {\frac {du}{-sin(x)}}\\[2ex]&=-\int 1-u^{2}du\\[2ex]&={\frac {4}{5}}[-u+{\frac {1}{3}}^{3}(x)]\\[2ex]&={\frac {4}{5}}[-cos(x)+{\frac {1}{3}}cos^{3}(x)]\\[2ex]&={\frac {4}{5}}[-cos({\frac {\pi }{2}})+{\frac {1}{3}}cos^{3}({\frac {\pi }{2}})-(-cos(0)+{\frac {1}{3}}cos^{3}(0))\\[2ex]&={\frac {4}{5}}[(-1)(0)+{\frac {1}{3}}(0)^{3}-((-1)(1)+{\frac {1}{3}}cos^{3}(1)\\[2ex]&={\frac {4}{5}}[-(-1+{\frac {1}{3}}(1)]\\[2ex]&={\frac {4}{5}}[{\frac {2}{3}}]\\[2ex]&={\frac {8}{15}}\\[2ex]\end{aligned}}$

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} }