Thank you in advance
Changing the "dummy" variable to a different letter, x, this is $\displaystyle \int_0^r xk_0\left(x\sqrt{\frac{s}{t}}\right)dx$. Now, the obvious first substitution is $\displaystyle y= x\sqrt{\frac{s}{t}}$ so $\displaystyle x= y\sqrt{\frac{t}{s}}$ and $\displaystyle dx= \sqrt{\frac{t}{s}}dy$. When $\displaystyle x= r$ $\displaystyle y= r\sqrt{\frac{s}{t}}$. The integral becomes $\displaystyle \frac{t}{s}\int_0^{r\sqrt{\frac{s}{t}}} yk_0(y)dy$. Now try a formula here: Bessel function of the first kind: Integration (subsection 21/01/02/01/01/01)