Fundamental theorem of Calculus

Question

Given that D(t) is a continuous and diffrentiable function .

Also D(0) = 0 , D(5) = 120 and D(20) = 200 , simplify the integral using fundamental theorem ,

∫ D'(t/4)dt inthe interval 0<t <20 ?

Answer 1

If the problem said ∫D'(t) from 0 to 20, you could take the antiderivative which is D(t) and substitute in 20 and 0. You would get D(20) - D(0). Since it says D(t/4) think about how you will have to adjust the result. You could use u-substitution to help figure it out.

Answer 2

Let t/4 = x, then t = 4x and dt = 4dx. and 0 ≤ x ≤ 5. So, ∫₀²⁰D'(t/4)dt = ∫₀⁵D'(x)4dx = 4D(x)₀⁵ = 4D(5) - 4D(0) =

= 4·120 - 4·0 = 480