How do I evaluate this definite integral?

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1 Answer
Ananda Dasgupta
Feb 5, 2018

#pi/12#

Explanation:

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

# 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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