Question Number 7184 by Yozzia last updated on 15/Aug/16
![Show that ∫(2^x /((1+(√5))^x +(3+(√5))^x ))dx=(1/(ln(1+(√5))−ln2))(ln[1+((((√5)−1)/2))^x ]−((((√5)−1)/2))^x )+C](Q7184.png)
$${Show}\:{that} \\ $$$$\int\frac{\mathrm{2}^{{x}} }{\left(\mathrm{1}+\sqrt{\mathrm{5}}\right)^{{x}} +\left(\mathrm{3}+\sqrt{\mathrm{5}}\right)^{{x}} }{dx}=\frac{\mathrm{1}}{{ln}\left(\mathrm{1}+\sqrt{\mathrm{5}}\right)−{ln}\mathrm{2}}\left({ln}\left[\mathrm{1}+\left(\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{2}}\right)^{{x}} \right]−\left(\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{2}}\right)^{{x}} \right)+{C} \\ $$$$ \\ $$