Evaluate the integral. (Remember the constant of integration.)

Question

∫2csc^4(x)cot^6(x)dx

Answer 1

!! I forgot the final parentheses on the final answer !! both fractions are multiplied by -2 in the final equation, sorry

Answer 2

First, we replace csc^4(x) = csc^2(x)*csc^2(x) = csc^2(x) (1 + cot^2(x)). This makes our integral become ∫2csc^4(x)cot^6(x)dx = ∫2csc^2(x) (1 + cot^2(x))cot^6(x)dx.

Recall that the derivative of cotangent is given by d/dx cot(x) = -csc^2(x). Using this we do a u-substitution with u = cot(x) and du = -csc^2(x).

The integral now becomes -∫2(1+u^2)u^6du = -∫2(u^6+u^8)du = -2u^9/9 - 2u^7/7 + C. Substituting for u, we have that ∫2csc^4(x)cot^6(x)dx = -2cot^9(x)/9 - 2cot^7(x)/7 + C