[tex]f(x) = \sqrt{2x-1}\\ \\ F(x) = \dfrac{2x-1}{3}\sqrt{2x-1} \\ \\ F(5) = \dfrac{9}{3}\cdot \sqrt{9} = 9\\ \\ F(1) =\dfrac{1}{3} \\ \\\displaystyle \int_{1}^5f(x)\cdot F^{2014}(x)\,dx =\int_{1}^5 F'(x)\cdot F^{2014}(x)\, dx = \\ \\ =\dfrac{F^{2015}(x)}{2015}\Big|_{1}^5 = \dfrac{F^{2015}(5)-F^{2015}(1)}{2015} = \dfrac{9^{2015}-\Big(\dfrac{1}{3}\Big)^{2015}}{2015} =[/tex]
[tex]= \Big(\dfrac{1}{3}\Big)^{2015}\cdot \dfrac{(9\cdot 3)^{2015}-1}{2015} = \dfrac{(3^{3})^{2015}-1}{2015\cdot 3^{2015}} = \dfrac{3^{6045}-1}{2015\cdot 3^{2015}}[/tex]
