Integration for this year

$\int_{0}^{\infty} \frac{2014^{-2013x} - 2014^{-2014x}}{x}$ Consider function, $f(a)= \int_{0}^{\infty} \frac{2014^{-ax} - 2014^{-2014x}}{x}$ $f(2014)=0$ . Now, differentiating under integration technique, just differentiate and integrate. We get $\log(\frac{2014}{2013})$ .

Otherwise, just use Frullani theorem.

Frullani's integral identity tells us that -

$\int_{0}^{\infty} \frac {f(ax)- f(bx)}{x} dx = (f(0) - \lim_{x \to \infty} f(x)) \cdot \ln \left(\frac {b}{a}\right)$

when the limit exists. Letting $f(x) = 2014^{-x}$ , the integral becomes -

$\int_{0}^{\infty} \frac {2014^{-2013x}-2014^{-2014x}} {x} dx = \ln\left(\frac {2014}{2013}\right)$