Difficult Indefinite Integral

Question

$$\int (\log x+1)x^x\,dx$$

This integral was found from the MIT Integration Bee. After making several unsuccessful attempts, I decided to type it into Mathematica, only to find that Mathematica could only produce an answer for this integral in the case where $\log(x)$ referred to the natural logarithmic function $\ln(x)$.

Answer 1

The form of the integrand suggests writing $$(1 + \log x)x^x = (1+\log x)e^{x \log x},$$ then observing that by the product rule, $$\frac{d}{dx}\bigl[x \log x\bigr] = x \cdot \frac{1}{x} + 1 \cdot \log x = 1 + \log x.$$ Consequently, the integrand is of the form $f'(x) e^{f(x)}$, and its antiderivative is simply $$e^{f(x)} = e^{x \log x} = x^x.$$

Answer 2

Use the substitution $y=x^x$, then do logarithmic differentitation. to get $(1+\ln x)x^x=\frac{dy}{dx}$. Now you may go from here.