Question

Assume that \(f\) is continuous everywhere and that \(c\) is a constant. Show that \(\int_{c a}^{c b} f(x) d x=c \int_{a}^{b} f(c x) d x\)

Short answer

By using variable substitution, we found that \(\int_{c a}^{c b} f(x) d x = c \int_{a}^{b} f(c x) dx\).

Step-by-step solution

Define the Variable Substitution

A variable substitution can simplify the integral. Let's use the substitution \(u = cx\). Then, the differential \(du = cdx\). Solving the differential for \(dx\) gives \(dx = du/c\).

Replace the Limits of Integration

With the new variable \(u\), the limits of integration change. Substituting \(u = cx\) into the limits, the lower limit \(ca\) becomes \(u = c^2 a\) and the upper limit \(cb\) becomes \(u = c^2 b\).

Carry Out the Substitution

Replace \(x\) and \(dx\) in the integral. The original integral \(\int_{c a}^{c b} f(x) dx\) becomes \(\int_{c^2 a}^{c^2 b} f(u/c) (du/c)\).

Simplify the Integral

Pull the constants out of the integral to simplify it. The integral becomes \(1/c \int_{c^2 a}^{c^2 b} f(u/c) du\).

Resubstitute the Original Limits of Integration

Return to the original limits of integration. Change the lower limit \(c^2 a\) to \(a\) and the upper limit \(c^2 b\) to \(b\). Our final form of the simplified integral is then \(c \int_{a}^{b} f(x) dx\).