Find the integral of 7 csc^3(x)/ (tanx) dx

Question

Find the integral of 7 csc³(x) / tan(x) dx

Answer 1

Since 1/tan(x) = cot x, the integrand is 7 csc²(x) * csc(x) cot(x); via the u-substitution u = csc x, du = -csc x cot x dx, we obtain

∫ 7 csc²(x) * csc(x) cot(x) dx = ∫ 7 u² (-du) = -7 ∫ u² du = -7u³/3 + C = -(7/3)csc³(x) + C

Answer 2