indefinite integrals

Question

can somebody help me with these problems?
gracias!!

1.∫csc3xcot3x dx

2.∫tan^7(x)sec^2(x) dx

3. ∫csc^9(x)cot(x) dx

thankssss (:

Answer 1

Use substitution

1. Let u=csc(3x), du=-3csc3x*cot3x dx
∫csc3xcot3x dx=∫(-1/3) du=-1/3 *u+c=-1/3 * csc(3x)+c

2. Let u=tanx, du=sec²x dx
∫tan^7(x) sec²x dx =∫u^7 du=(1/8)*u^8+c=(1/8)*tan^8(x)+c

3. Let u=cscx, du=-cscx*cotx dx
∫csc^9(x)cotx dx=-∫u^8 du=-1/9 * csc^9(x) +c