Evaluate the Trig integral. Show All Steps

Question

Evaluate the Trig integral. Show All Steps. If possible can you attach a video going through it.

Answer 1

Recall that 1 + cot²x = csc²x

∫cot⁵x csc³x dx = ∫cot⁴x csc²x cotx cscx dx = ∫ (csc²x - 1)² csc²x cotx cscxdx

Let u = cscx, then du = -cscxcotxdx.

So, we have -∫ (u² - 1)² u² du = -∫(u⁴ - 2u² + 1)u²du = -∫ (u⁶ - 2u⁴ + u²)du

= -(1/7)csc⁷x + (2/5)csc⁵x - (1/3)csc³x + C

Evaluate from π/4 to π/2: (-1/7 + 2/5 - 1/3) - (-8√2/7 + 8√2/5 - 2√2/3)

= -8/105 - (-22√2/105) = (22√2 - 8) / 105 ≈ 0.22

Answer 2

Hint: Write everything in terms of cosine and sine. After that, recall that cos²(x) + sin²(x) = 1. Then, use the u-substitution. Let me know if you still need help.

Answer 3