# Integral of $e^{\frac{y}{x}}$

How can we evaluate the following?

$$\int e^{\frac{y}{x}}\ \mathrm dy$$

The answer I got is $x e^{y/x}$.

• With regards to which variable are you integrating? $x$? $y$? Both? If so, in which order? – daOnlyBG Nov 20 '14 at 16:07
• In any event, you forgot the constant! – GFauxPas Nov 20 '14 at 16:08
• integrating with respect to dy – Vaquita Nov 20 '14 at 16:10
• Change variable $y=z x$ and it will becopme easy. – Claude Leibovici Nov 20 '14 at 16:32
• How did you get that answer? If you don't show the steps, we can't tell if they are correct. – robjohn Nov 20 '14 at 17:11

\begin{align} \int_{- \infty} e^{\frac{y}{x}}dy &= \int_{- \infty}^y e^{\frac{t}{x}}dt \\ &= \int_{- \infty}^y e^{\frac{xu}{x}}d(xu) \\ &= x \int_{- \infty}^{y \over x} e^u d u \\ &= x \vert_{u = {- \infty}}^{y \over x} e^u \\ &= x \ e^{y \over x} \end{align}
First note that $$\int a^t\ \mathrm dt=\frac{a^t}{\ln a}+C$$ Therefore $$\int e^{\frac{y}{x}}\ \mathrm dy=\int \left(e^{\frac{1}{x}}\right)^y\ \mathrm dy$$ $$=\frac{\left(e^{\frac{1}{x}}\right)^y}{\ln e^{\frac{1}{x}}}+C=xe^{\frac{y}{x}}+C$$