How can i integrate this function?

Question

How can i integrate this? I tried by parts but it didn't end well. $$\int_0^\infty \frac{e^{-5x}-e^{-10x}} x \, dx$$

Answer 1

Observe we have that \begin{align} \frac{e^{-5x}-e^{-10x}}{x} = \int^{10}_{5} \frac{e^{-tx}}{x}\ dt\sim -\int^{10}_5e^{-tx}\ dt \end{align} which means \begin{align} \int^\infty_0\frac{e^{-5x}-e^{-10x}}{x}\ dx =&\ -\int^\infty_0\int^{10}_{5}e^{-tx}\ dtdx\\ =&\ \int^{10}_{5}\int^\infty_0e^{-tx}dxdt = \int^{10}_5 \frac{1}{t}\ dt \\ =&\ \log 10-\log 5= \log 2. \end{align}