Change of variables for an integral

Question

I have an integral where I need to change variables. The integral has the form,

$\int_0^x f(x,t) dt$ .

I change variables/rescale by setting $\tilde{t}=xt$, which means $d\tilde{t}=xdt$. Would the new integral have the following form & bounds:

$\int_0^{x^2} f(x,\tilde{t}) d\tilde{t}$ ?

Answer 1

As you correctly computed, $d\widetilde{t}=xdt$, so $dt=d\widetilde{t}/x$. We can see by $\widetilde{t}=xt$ that $0\le t\le x$ means that $0\le \widetilde{t}\le x^2$ using $\widetilde{t}/x=t$. Plugging this all in, we get $$ \int_0^x f(x,t)dt=\int_0^{x^2} \frac{f(x,\widetilde{t}/x)}{x}d\widetilde{t}.$$