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)-2+4 = -2\ln(2)+2}

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{\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{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_{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} }

${\begin{aligned}-\int ({\frac {1}{u}})dx&=-\ln |u|+C&=\ln |\cos(x)^{-1}|+C&=\ln |{\frac {1}{cos(x)}}|+C&=\ln |\sec(x)|+C\end{aligned}}$