Question

How do you integrate `int (9+x^2)/sqrt(4 - x^2)dx` using trigonometric substitution?

Answer 1

`11sin^(-1)(x/2) + 4 sin(2sin^(-1)(x/2))`

Explanation:

`int (9 + x^2)/sqrt(4-x^2)`
Sub x=2 sin `theta`
`dx/d(theta)`=2cos`theta`
`int(9+4(sin theta))^2/sqrt(4(cos theta)^2)` x 2cos`theta`
=`int 9+4 (sin theta)^2`
= `int 9 + 4 times 1/2(1-cos2 theta)`

--> `cos2 theta = (cos theta)^2-(sin theta)^2`
--> `cos 2 theta = 1-2(sin theta)^2`
--> `(sin theta)^2 = 1/2(1-cos2 theta)`

= `int 9 +2-2cos2theta`
= `int 11-2cos2 theta`
= `11theta+4sin 2theta`
= `11sin^(-1)(x/2) + 4 sin(2sin^(-1)(x/2))`