Question

How to solve this integration problem? the answer should be -π/12

Answer 1

`\displaystyle\int_(-1)^(-\sqrt2/2)(\mathrm dy)/(y√(4y^2-1))`

Try setting
`y=\frac12\sec t`, so that
`\mathrm dy=\frac12\sec t\tan t\,\mathrm dt`. The interval then changes from
`-1\le y\le-\frac{\sqrt2}2` to
`\frac{2\pi}3\le t\le \frac{3\pi}4`.

Now, the integral is


`\displaystyle \int_(2\pi/3)^(3\pi/4) (\frac12\sec t\tan t)/(\frac12\sec t√(4\left(\frac12\sec t\right)^2-1))\,\mathrm dt=\int_(2\pi/3)^(3\pi/4)(\tan t)/(√(\sec^2t-1))\,\mathrm dt`

The Pythagorean identity lets you reduce the denominator:


`√(\sec^2t-1)=√(\tan^2t)=|\tan t|`

Since
`\tan t<0` for the given interval, you have
`|\tan t|=-\tan t`, which means the integral is equal to


`\displaystyle\int_(2\pi/3)^(3\pi/4)(\tan t)/(-\tan t)\,\mathrm dt=-\int_(2\pi/3)^(3\pi/4)\mathrm dt=-\frac\pi{12}`