tan sec Power Integral Problem

[Jason76]

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5 \tan^{5}(x) \sec^{4}(x) dx\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[\tan^{5}(x))(\sec^{2}{x})(\sec^{2}(x))\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}}5[\tan^{5}(x))(1 + tan^{2}(x))(\sec^{2}(x))\)

\(\displaystyle u = \tan x\)

\(\displaystyle du = \sec^{2}(x) dx\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[(u^{5})(1 + u^{2})]\)

\(\displaystyle \dfrac{5u^{6}}{6} + \dfrac{u^{11}}{11}\) evaluated at \(\displaystyle [0,\dfrac{\pi}{3}\)

\(\displaystyle [\dfrac{5\tan^{6}(x)}{6} + \dfrac{\tan^{11}(x)}{11}]\)evaluated at \(\displaystyle [0,\dfrac{\pi}{3}]\)

\(\displaystyle [\dfrac{5\tan^{6}(\dfrac{\pi}{3})}{6} + \dfrac{\tan^{11}(\dfrac{\pi}{3})}{11}] - [\dfrac{5\tan^{6}(0)}{6} + \dfrac{\tan^{11}((0)}{11}] = 60.76257693\) ??

 

[Deleted member 4993]

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5 \tan^{5}(x) \sec^{4}(x) dx\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[\tan^{5}(x))(\sec^{2}{x})(\sec^{2}(x))\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}}5[\tan^{5}(x))(1 + tan^{2}(x))(\sec^{2}(x))\)

\(\displaystyle u = \tan x\)

\(\displaystyle du = \sec^{2}(x) dx\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[(u^{5})(1 + u^{2})]\) = \(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[(u^{5} + u^{7})]du\)

\(\displaystyle \dfrac{5u^{6}}{6} + \dfrac{u^{11}}{11}\) incorrect....evaluated at \(\displaystyle [0,\dfrac{\pi}{3}\)

\(\displaystyle [\dfrac{5\tan^{6}(x)}{6} + \dfrac{\tan^{11}(x)}{11}]\)evaluated at \(\displaystyle [0,\dfrac{\pi}{3}]\)

\(\displaystyle [\dfrac{5\tan^{6}(\dfrac{\pi}{3})}{6} + \dfrac{\tan^{11}(\dfrac{\pi}{3})}{11}] - [\dfrac{5\tan^{6}(0)}{6} + \dfrac{\tan^{11}((0)}{11}] = 60.76257693\) ??

Review the power-laws!!

 

[Jason76]

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5 \tan^{5}(x) \sec^{4}(x) dx\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[\tan^{5}(x))(\sec^{2}{x})(\sec^{2}(x))\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}}5[\tan^{5}(x))(1 + tan^{2}(x))(\sec^{2}(x))\)

\(\displaystyle u = \tan x\)

\(\displaystyle du = \sec^{2}(x) dx\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[(u^{5})(1 + u^{2})] du\)


\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5[(u^{5}) + u^{7}) ]du\)

\(\displaystyle \int_{0}^{\frac{\pi}{3}} 5u^{5}) + 5u^{7}) ]du \)

\(\displaystyle = 5\dfrac{u^{6}}{6} + 5\dfrac{u^{8}}{8}] du\)

\(\displaystyle = 5\dfrac{\tan^{6}(x)}{6} + 5\dfrac{\tan^{8}(x)}{8}] \) evaluated at 0 and \(\displaystyle \pi/3\)

\(\displaystyle = 5\dfrac{\tan^{6}(\pi/3)}{6} + 5\dfrac{\tan^{8}(\pi/3)}{8}] - 5[\dfrac{\tan^{6}(0)}{6} + 5\dfrac{\tan^{8}(0)}{8}] = \dfrac{585}{5} \)

 

[Jason76]

Misreading of final answer on calculator

Correct answer: 585/8