indefinite integral of (tan^2(x))^1/3sec^2(x)dx

Question

Find the indefinite integral and simplify your result. Please show step-by-step result.

Answer 1

Find F(x) = ∫(tan^2(x))^(1/3)sec^2(x)dx.

u = tan^2(x) ==> tan(x) = u^(1/2)

du = 2tan(x)sec^2(x)dx

sec^2(x)dx = 1/2 u^(-1/2)du

F(u) = 1/2 ∫u^(1/3)u^(-1/2)du

= 1/2 ∫u^(-1/6)du

= 1/2 u^(-1/6+1)/(-1/6+1)

= 1/2 u^(5/6)/(5/6)

= (1/2)(6/5) u^(5/6)

= (3/5) u^(5/6)

F(x) = (3/5) (tan^2(x))^(5/6) + C

F(x) = (3/5) (tan(x))^(5/3) + C