Question

How do you integrate from -6 to-3 for `sqrt((x^2-9))/2`?

Answer 1

Method is this

Substitute `x= 3 sec theta`
For any variable substitution in the integration, Integrant, Limits and the integration operator need to be changed.

So Integrant is `sqrt(9sec^2theta-9)/2 = (3tantheta)/2`

The limits are `@x=-6 to x=-3` changes as well.

Upper Limit calculation
`3sectheta = -3`
`sec theta = -1`
`theta = pi`

Lower Limit calculation
`3sec theta = -6`
`sec theta = -2`
`theta = (5pi)/6`

Integral operator change
`dx = sectheta tantheta d theta`
`int_-6^-3 sqrt(x^2-9)/2 = int_((5pi)/6)^pi (3secthetatan^2theta d theta)/2`
`= 3/2int_((5pi)/6)^pi (sec^3theta - sec theta) d theta`

Check how to evaluate `int sec^3 theta d theta` here

`=3/2{1/2(sec theta tantheta+ln|sec theta+tantheta|)-ln|sec theta+tantheta|}_((5pi)/6)^pi`

`=3/4{sec theta tantheta-ln|sec theta+tantheta|}_((5pi)/6)^pi`
`=-0.0082`