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How to find indefinite integral $$\int a^{\frac {1}{x}} \mathrm dx$$?

The problem is that we can't use the formula for $a^x$.

Any help welcomed

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It is non-elementary. –  L. F. Feb 9 '13 at 13:53
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That is, there is no "nice" closed form. To find the antiderivative, one would have to resort to infinite series reprensentations and the like... (A substitution and integrating by parts allows one to write the indefinite integral in terms of the Exponential integral.) –  David Mitra Feb 9 '13 at 13:58
    
is it possible to circumvent exponential integral? –  David Hoffman Feb 9 '13 at 14:01

1 Answer 1

Let $t = \log(a)$. Make $u$-substitution $u=\frac{1}{x}$: $$ \int \exp\left(\frac{t}{x}\right) \mathrm{d}x = \int \exp\left( t u \right) \mathrm{d} \left(\frac{1}{u}\right) \stackrel{\text{by parts}}{=} \frac{\exp(t u)}{u} - t \int \frac{\exp(t u)}{u} \mathrm{d} u \stackrel{u \to u/t}{=} \left( \frac{\exp(t u}{u} - t \int \frac{\exp(u)}{u} \mathrm{d} u \right) $$ The latter integral is non-elementary. A special function exponential integral $\operatorname{Ei}(u)$ has the require antiderivative.

Answers to the earlier question is a recommended read.

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