6.1 Areas Between Curves/15
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} & \color{red}\mathbf{y=\tan(x)} & \color{royalblue}\mathbf{y= 2\sin(x)} \\ & x=-\frac{\pi}{3} & x=\frac{\pi}{3} \\ \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_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left[(\tan(x)) - (2\sin(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 \begin{align} \tan(x) &= 2\sin(x) \\ \tan(x)-2\sin(x) &= 0 \\ x &= 0 \\ \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_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left[(\tan(x)) - (2\sin(x))\right]dx = \int_{-\frac{\pi}{3}}^{0}\left[(\tan(x)) - (2\sin(x))\right]dx + \int_{0}^{\frac{\pi}{3}} \left[(2\sin(x)) - (\tan(x))\right]dx = -2\ln(2)-1-1-\ln(2)+2 = 2-\ln(2)-1-1+4 = -2\ln(2)+2}
![{\displaystyle {\begin{aligned}\int _{-{\frac {\pi }{3}}}^{0}\left[(\tan(x))-(2\sin(x))\right]dx\\[2ex]&=\left[\ln |\sec(x)|+2\cos(x)\right]{\Bigg |}_{-{\frac {\pi }{3}}}^{0}\\[2ex]&=\left[\ln |\sec(0)|+2\cos(0)\right]-\left[\ln |\sec(-{\frac {\pi }{3}})+2\cos(-{\frac {\pi }{3}})|\right]\\[2ex]&=\left[0+2\right]-\left[\ln(2)-2({\frac {1}{2}})\right]=2-\ln(2)-1\\[2ex]&=2-\ln(2)-1\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/69bf9719295c930e76f7ab641e003c5313eb4188)
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_{0}^{\frac{\pi}{3}} \left[(2\sin(x)) - (\tan(x))\right]dx \\[2ex] &= \left[-2\cos(x)-\ln|\sec(x)|\right]\Bigg|_{0}^{\frac{\pi}{3}} \\[2ex] &= \left[-2\cos(\frac{\pi}{3})-\ln|\sec(\frac{\pi}{3})|\right] + \left[2\cos(0)+\ln|\sec(0)|\right] \\[2ex] &= \left[(-2)(1/2)-\ln(2)\right]+\left[2+0\right] = -1-\ln(2)+2 \\[2ex] &= -1-\ln(2)+2 \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 \text{Note: } \int\tan(x)dx=\int\frac{\sin(x)}{\cos(x)}dx=\ln|\sec(x)|+C}
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 &= \cos(x) \\[2ex] du &= -\sin(x) \\[2ex] -du &= \sin(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} -\int (\frac{1}{u})dx &= -\ln|u|+C &=\ln|\cos(x)^{-1}|+C &=\ln|\frac{1}{cos(x)}|+C &=\ln|\sec(x)|+C \end{align} }