7.1 Integration By Parts/50

Prove $\int _{}^{}\sec ^{n}(x)dx={\frac {\tan(x)\cdot \sec ^{n-2}(x)}{n-1}}+{\frac {n-2}{n-1}}\int _{}^{}\sec ^{n-2}(x)dx$

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_{}^{} \sec^{n}(x)dx = \int_{}^{} \sec^{2}(x) \cdot \sec^{n-2}(x) 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 \begin{align} &u = \sec^{n-2}(x) \quad dv= \sec^{2}(x)dx \\[2ex] &du = (n-2)\sec^{n-3} \cdot \sec(x) \tan(x) dx \quad v= \tan(x) \\[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_{}^{} \sec^{2}(x) \cdot \sec^{n-2}(x) dx &= \sec^{n-2}(x) \cdot \tan(x) - \int_{}^{} \left[(n-2)\sec^{n-3}(x) \cdot \sec(x)\tan(x)\right]\cdot \tan(x)dx \\[2ex] &= \sec^{n-2}(x) \cdot \tan(x) - (n-2)\int_{}^{} \left[\sec^{n-2}(x) \cdot \tan^{2}(x)\right]dx \\[2ex] &= \sec^{n-2}(x) \cdot \tan(x) - (n-2)\int_{}^{} \sec^{n-2}(x) \cdot [\sec^{2}(x)-1]dx \\[2ex] &= \sec^{n-2}(x) \cdot \tan(x) - (n-2)\int_{}^{} \left[\sec^{n}(x) - \sec^{n-2}(x)\right]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_{}^{} \sec^{n}(x)dx = \sec^{n-2}(x) \cdot \tan(x) - (n-2)\int_{}^{} \sec^{n}(x)dx + (n-2) \int_{}^{}\sec^{n-2}(x)dx } \end{align}