Question

100 POINTS.

PLEASE PROVIDE STEPS

Answer 1

12 ∫ (cot⁴(3x) csc²(3x)) dx

If u = cot(3x), then du = -3 csc²(3x) dx. So -⅓ du = csc²(3x) dx.

12 ∫ u⁴ (-⅓ du)

-4 ∫ u⁴ du

-⅘ u⁵ + C

Substituting back:

-⅘ cot⁵(3x) + C

Evaluate between x=0 and x=π/12. cot(0) is undefined, so the integral does not exist.

Explanation:

Answer 2

Explanation:

12 ∫ (cot⁴(3x) csc²(3x)) dx

If u = cot(3x), then du = -3 csc²(3x) dx. So -⅓ du = csc²(3x) dx.

12 ∫ u⁴ (-⅓ du)

-4 ∫ u⁴ du

-⅘ u⁵ + C

Substituting back:

-⅘ cot⁵(3x) + C

Evaluate between x=0 and x=π/12. cot(0) is undefined, so the integral does not exist.