If f is continuous and ∫0 to 8 f(x)dx=10, evaluate the definite integral. ∫ 0 to 2 f(4x)dx?

Question

If f is continuous and ∫0 to 8 f(x)dx=10, evaluate the definite integral. ∫ 0 to 2 f(4x)dx?

 

please help

Answer 1

If you use the substitution method and the fundamental theorem of calculus you can get the following:

U = 4x, dU = 4dx or dx = (1/4)dU

now the limits of integration for U are found by plugging in the limits to our formula for U, so our limits are U = 4*0 = 0 and U = 4*2 = 8

plugging all of this into the integral we get

(integral from 0 to 8): f(U)(1/4)dU

The constant just pulls out front and then we have the same integral as before F(8)-F(0) = 10, Where F(x) is the antiderivative of f(x) (you can interchange x with whatever variable you like)

so our final answer is (1/4)(F(8) - F(0)) = 5/2