What is \(\displaystyle\int_{1}^{2}x^{x}(1+\ln x)\,dx\) equal to?
3
Notice that \(\dfrac{d}{dx}\left(x^{x}\right)=x^{x}(1+\ln x)\), so the given integrand is exactly the derivative of \(x^{x}\). Hence \(\int_{1}^{2}x^{x}(1+\ln x)\,dx=\Big[x^{x}\Big]_{1}^{2}=2^{2}-1^{1}=4-1=3\).
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