Question

Evaluate the integral: csc²x(cotx-1)³ dx

Answer 1

 -(cotx-1)⁴/4 + C

Explanation:

Let u = cotx - 1. Then du = -csc²x dx.

Substituting u and du into the integral, we get:

∫ csc²x(cotx-1)³ dx = -∫ u³ du

Now, we can evaluate the integral using the reverse power rule:

∫ uⁿ du = u^(n+1)/(n+1) + C

Put n = 3

-∫ u³ du = -u⁴/4 + C

substituting u back to cotx - 1

-u⁴/4 + C = -(cotx-1)⁴/4 + C

Therefore, the value of the integral is -(cotx-1)⁴/4 + C.