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 \\[2ex] }

Note: 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} \tan^{2}(x) = \sec^{2}(x)-1 \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 \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}(x)dx }

Solving for 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^{2}x)(\tan^{n-2}x) }

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

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

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

Bring down:

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_{}^{}\tan^{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} = \frac{\tan^{n-1}(x)}{n-1} -\int_{}^{}\tan^{n-2}(x)dx \end{align} }