If 𝑓 is continuous and....

Question

If f is continuous and

 

∫ (1 to 8) f(t) dt = 10,

find

∫ (1 to 2) t^2f(t^3) dt

Answer 1

U = t^3

dU = 3* t^2 dt

(1/3) du = t^2 dt

integral [f(t^3) * t^2 dt] , t=0....2

= (1/3) integral [ f(U) du ], U = 0...8

= 10/3

Answer 2

Let u = t³. Then, du = 3t²dt. So, t²dt = (1/3)du.

When t = 1, u = 1³ = 1 and when t = 2, u = 2³ = 8.

So, ∫_{(from 1 to 2)} t² f(t³) dt = (1/3)∫_{(from 1 to 8)} f(u) du = (1/3)(10) = 10 / 3