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Math Help - evaluate an integral

  1. #1
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    evaluate an integral

    How do I compute this integral?

    \int \frac{e^\sqrt5x}{3x} dx
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  2. #2
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    Re: evaluate an integral

    Quote Originally Posted by dumbledore View Post
    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*}?
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  3. #3
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    Re: evaluate an integral

    Second one - sorry about that
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    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?
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  5. #5
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    Re: evaluate an integral

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