Show, that if $f$ is a continuous function and $a < b$, then the integral $\int_{a}^{b}f(x)dx $ can be transformed with a substitution $x=At+B$ to an integral over the interval $[\alpha, \beta]$. Find the parameters $A$ and $B$ and perform the substitution.
Hello guys, I had this question in my exam but I wasn't able to understand this clearly. I tried substituting $x=At+B$ in the given integral and get $\frac{1}{A}\int_{\frac{a-B}{A}}^{\frac{b-B}{A}}f(At+B).d(At+B)$. And then what am I supposed to do next? Do I just put $\frac{b-B}{A} = \beta$ and $\frac{a-B}{A} = \alpha$ and then solve for A and B?
Can someone please give me a hint on how I should do this problem? Thank you so much for your help!
I think you can use:
$a = A \alpha + B, \quad b = A \beta + B.$
So $$A = \frac{a-b}{\alpha - \beta}, \quad B = \frac{-a\beta+b\alpha}{\alpha-\beta},$$
and the integral is: $$\frac{a-b}{\alpha-\beta}\int_\alpha^\beta f\left(\frac{a-b}{\alpha-\beta}t+\frac{-a\beta+b\alpha}{\alpha-\beta}\right)dt.$$