7.1 Integration By Parts/49

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

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) }

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 = \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{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)(\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{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_{}^{}(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} }