Is there any way to change $\int\limits_0^c {f\left( x \right){e^{ - ax}}dx} \to \int\limits_0^\infty {f\left( u \right){e^{ - au}}du}$?

Question

Is there any way to make a change of variable such that ?

$\int\limits_0^c {f\left( x \right){e^{ - ax}}dx} \to \int\limits_0^\infty {f\left( u \right){e^{ - au}}du}$

Thank you very much !

Answer 1

$$\int_0^{\infty} f(u) e^{-au} du = \int_0^{c} f(u) e^{-au} du \ + \int_c^{\infty} f(u)e^{-au} du$$

What you want is only possible, hence, when $$\int_c^{\infty} f(u) e^{-au} du =0$$