Question

Trig substitutions..

Answer 1

let x = 2 sec (theta) ----> x^3 = 8 sec^3 (theta)

Answer 2

x^2 = 4 sec^2 (theta). minus both sides by 4
x^2 -4 = 4 (sec^2(theta) -1) = 4 tan^2 ( theta)

Answer 3

change the limit: when x = 2sqr(2), it means 2 sqr(2) = 2 sec (theta)---> cos (theta) = sqr(2)/2 ---> theta = pi/4
when x =4, the same process to get theta = pi/3

Answer 4

dx = 2 sec(theta) tan(theta) dtheta

Answer 5

replace every thing into the original function we get: \[\int\limits_{\pi/4}^{\pi/3} \frac{2 \sec \theta *\tan \theta }{16\sec^3\theta*\tan \theta }d \theta \]

Answer 6

I think you can handle the leftover.
hope this help