Question

How do I evaluate this definite integral?

Answer 1

`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`