How do I evaluate this definite integral?

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How do I evaluate this definite integral?

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Answer 1 · 2018-02-05T04:53:54

$\frac{π}{12}$ $pi/12$

Explanation:

Try the substitution $3 x = sec θ$ $3x = sec theta$ so that
$3 d x = sec θ tan θ d θ$ $3dx = sec theta tan theta d theta$ . Also the upper limit becomes
${sec}^{- 1} (2) = {cos}^{- 1} (\frac{1}{2}) = \frac{π}{3}$ $sec^{-1} (2) = cos^{-1} (1/2) = pi/3$ and the lower limit becomes ${sec}^{- 1} (\sqrt{2}) = {cos}^{- 1} (\frac{1}{\sqrt{2}}) = \frac{π}{4}$ $sec^{-1} (sqrt{2}) = cos^{-1} (1/{sqrt{2}}) = pi/4$
The integral is

$\int_{\frac{π}{4}}^{\frac{π}{3}} \frac{sec θ tan θ d θ}{sec θ tan θ} = \int_{\frac{π}{4}}^{\frac{π}{3}} d θ = \frac{π}{3} - \frac{π}{4} = \frac{π}{12}$ $int_{pi/4}^{pi/3}{ sec theta tan theta d theta}/{sec theta tan theta} = int_{pi/4}^{pi/3} d theta = pi/3-pi/4 = pi/12$

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How do I evaluate this definite integral?

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