# Evaluation of $\int_{0}^{1} (\frac{1}{x}) ^{\log x}\,\mathrm dx$ which has a nice closed form

I am interested to evaluate integral of the form $\displaystyle\int {g(x)}^{g'(x)}\,\mathrm dx$. I have got this simple example: $\displaystyle\int_{0}^{1} \left(\frac{1}{x}\right) ^{\log x}\,\mathrm dx$. Wolfram Alpha gives a nice closed form which is defined as shown below: $$\int_0^1\left(\dfrac1x\right)^{\log(x)}\,\mathrm dx = -\dfrac12\sqrt[4]{e}\sqrt\pi\left(\mathrm{erf}\left(\dfrac12\right) - 1\right) \approx 0.545641\tag{1}$$

Now my question here is: Is there any mathematical basis that gives us rule(s) to evaluate integrals of the form $\displaystyle\int {g(x)}^{g'(x)}\,\mathrm dx$? And at the same time how can we arrive at the solution in $(1)$?

• Change $t=\log x$ works for the identity. Not clear what $g^{g'}$ is. For the example, it is $(g')^g$ if $g'$ is the derivative. – A.Γ. May 27 '18 at 22:08
• Very easy, just change of variables and completing the square. – Szeto May 27 '18 at 22:33