7.1 Integration By Parts/50

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_{}^{} \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 }

${\begin{aligned}&u=\sec ^{n-2}(x)\quad &dv=\sec ^{2}(x)dx\\[2ex]&du=(n-2)\sec ^{n-3}\cdot \sec(x)\tan(x)dx\quad &v=\tan(x)\\[2ex]\end{aligned}}$

${\begin{aligned}\int _{}^{}\sec ^{2}(x)\cdot \sec ^{n-2}(x)dx&=\sec ^{n-2}(x)\cdot \tan(x)-\int _{}^{}\left[(n-2)\sec ^{n-3}(x)\cdot \sec(x)\tan(x)\right]\cdot \tan(x)dx\\[2ex]&=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\left[\sec ^{n-2}(x)\cdot \tan ^{2}(x)\right]dx\\[2ex]&=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\sec ^{n-2}(x)\cdot [\sec ^{2}(x)-1]dx\\[2ex]&=\sec ^{n-2}(x)\cdot \tan(x)-(n-2)\int _{}^{}\left[\sec ^{n}(x)-\sec ^{n-2}(x)\right]dx\\[2ex]\end{aligned}}$

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_{}^{} \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} }

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} &+(n-2)\int_{}^{} \sec^{2}(x)dx \quad &&&+(n-2)\int_{}^{} \sec^{2}(x)dx \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} = \sec^{2}(x)\tan(x) + (n-2) \int_{}^{} \sec^{n-2}(x)dx = \frac{\sec^{n-2}(x) \tan(x)}{n-1} + \frac{n-2}{n-1} \int_{}^{} \sec^{n-2}(x)dx \end{align} }