How to integrate $\dfrac1{5^{\ln(x)}}$?

Question

$\int\dfrac1{5^{\ln(x)}}{\rm d}x$, how to start? Just a little hint please.

Answer 1

Hint: $$5^{\ln x} = \exp(\ln( 5^{\ln x})) = \exp(\ln(x) \ln 5) = x^{\ln 5}.$$

Answer 2

$$ \frac{1}{5^{\ln x}} = 5^{-\ln x} = \exp(\ln(5^{-\ln x})) = \exp(-\ln(x)\ln(5))) = \exp(\ln(x^{-\ln(5)}) = x^{-\ln(5)}. $$

Answer 3

Let $y=\dfrac1{5^{\ln x}}$

$\ln y=-\ln5\ln x$

$-\dfrac{\ln5}x=\dfrac{d(\ln y)}{dx}=\dfrac{d(\ln y)}{dy}\dfrac{dy}{dx}=\dfrac1y\dfrac{dy}{dx}$