calculate the indefinite integral

2sec^2x dx

(cosx)/(sin^3x) dx

calculate the definite integral
interval pi/4, pi/12

csc2xcot2x dx

Well, I happen to know that the derivate of the tangent is sec^2 :)

(cosx)/(sin^3x) dx

Try f(x) = 1/sin^2 x
f' = - 2 sin x cos x /sin^4 x
see what happens?

let u = 2 x
then du = 2 dx
and we have
csc u cot u du/2 from u = pi/2 to pi/6

(1/sin u)(cos u/sin u) du/2

= (1/2) (cos u/sin^2 u) du

try 1/sin u