1. ## evaluate an integral

How do I compute this integral?

$\int \frac{e^\sqrt5x}{3x} dx$

2. ## Re: evaluate an integral

Originally Posted by dumbledore
How do I compute this integral?

$\int \frac{e^\sqrt5x}{3x} dx$
Is it \displaystyle \begin{align*} \int{\frac{e^{\sqrt{5}\,x}}{3x}\,dx} \end{align*} or \displaystyle \begin{align*} \int{\frac{e^{\sqrt{5x}}}{3x}\,dx} \end{align*}?

3. ## Re: evaluate an integral

Second one - sorry about that

4. ## Re: evaluate an integral

\displaystyle \begin{align*} \int{\frac{e^{\sqrt{5x}}}{3x}\,dx} &= \frac{1}{3} \int{ \frac{e^{\sqrt{5}\,\sqrt{x}}}{\sqrt{x}\,\sqrt{x}} \, dx } \\ &= \frac{ 2 }{ 3 \, \sqrt{5} } \int{ \frac{ \sqrt{5} \, e^{ \sqrt{5} \, \sqrt{x} } }{ 2 \, \sqrt{x} \, \sqrt{x} } \,dx } \end{align*}

Now make the substitution \displaystyle \begin{align*} u = \sqrt{5} \, \sqrt{x} \implies du = \frac{ \sqrt{5} }{ 2 \, \sqrt{x} } \, dx \end{align*} and the integral becomes

\displaystyle \begin{align*} \frac{ 2 }{ 3 \, \sqrt{5} } \int{ \frac{ \sqrt{5} \, e^{ \sqrt{5} \, \sqrt{x} } }{ 2 \, \sqrt{x} \, \sqrt{x} } \,dx } &= \frac{ 2 }{ 3 \, \sqrt{5} } \int{ \frac{ e^u }{ \frac{ u }{ \sqrt{5} } } \, du } \\ &= \frac{2}{3} \int{ \frac{e^u}{u} \,du} \end{align*}

and this can only be evaluated in terms of the Exponential Integral. Have you encountered this before?

5. ## Re: evaluate an integral

No I have not encountered this before. Is there any other way of solving the integral?