Find definite integral if f(x) is continuous and...

Question

If f(x) is continuous and definite integral ∫ [b=8 and a=4] f(x)dx=32

Then find

Definite integral ∫ [b=2, a=1] f(4x) dx

Answer 1

Let's say ∫f(x)dx = F(x) + C if f(x) is continuous, then ∫^b_a f(x)dx = F(b) - F(a)

Given:

∫⁸₄ f(x)dx = F(8) - F(4) = 32

Evaluate:

∫²₁ f(4x)dx

u = 4x

du = 4dx

(1/4)du = dx

----

u=4 when x=1, and u=8 when x=2

substitute:

(1/4)∫⁸₄ f(u)du = (1/4)[F(8) - F(4)] = (1/4)(32) = 8