Uses of integral formula $\int_{ca}^{cb}f(t)dt=c\int_{a}^{b}f(ct)dt$

Question

I came across this formula working through Spivak for the first time. The book is a treasure trove of neat and seemingly very useful facts like this.

I was wondering if people had some favorite uses of this fact to evaluate some definite integrals.

In Spivak, one of the subsequent problems has you use the fact that if the area of the unit circle is $\pi$ then the area of the ellipse $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$ is $\pi ab$ which I did as follows. Given $$ 2\int_{-1}^{1}\sqrt{1-t^2}dt=\pi $$ And combining this with the integral for the area of an ellipse $$ b\int_{-a}^{a}\sqrt{1-\frac{t^2}{a^2}}dt=ab\int_{-1}^{1}\sqrt{1-t^2}dt=ab\pi $$

Thanks!

Answer 1

\begin{gather*} \log(xy) = \int_1^{xy}\frac{dt}{t} \\ = \int_1^{x}\frac{dt}{t} + \int_x^{xy}\frac{dt}{t} \\ = \int_1^{x}\frac{dt}{t} + \int_1^{y}\frac{du}{u} \\ = \log x + \log y. \end{gather*}