7.1 Integration By Parts/49

Prove 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 \int_{}^{} \left(\tan^{n}(x)\right)dx =\frac{\tan^{n-1}x}{n-1} - \int_{}^{} \left(\tan^{n-2}x\right)dx }

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 \int_{}^{} \left(\tan^{n}(x)\right)dx = \int_{}^{} \left((\tan^{2}x)(\tan^{n-2}x)\right)dx = \int_{}^{} (\sec^{2}(x)-1)\tan^{n-2}(x) dx = \int_{}^{} (\sec^{2}x)(\tan^{n-2}x)-\tan^{n-2}xdx = \int_{}^{} (\sec^{2}x)(\tan^{n-2}x) -\int_{}^{}\tan^{n-2}xdx }

Solving for $\int _{}^{}(\sec ^{2}x)(\tan ^{n-2}x)$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}&u=\tan ^{n-2}x\quad dv=1dx\\[2ex]&du=1dx\quad v=x\\[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_{}^{} \left(\ln(x)^{n}\right)dx &= x \ln(x)^{n} - \int_{}^{} \left((x \frac{n \ln(x)^{n-1}}{x}) \right)dx \\[2ex] &= x \ln(x)^{n} - \int_{}^{} \left(n \ln(x)^{n-1} \right)dx \\[2ex] &= x \ln(x)^{n} - n \int_{}^{} \left(\ln(x)^{n-1} \right)dx \\[2ex] \end{align} }