Question

Solve the given expression

Answer 1

$let\ x-\frac{10}{8}=t^2\\\\dx=2t\ dt,\ \ \ x=t^2+1.25\\\\\int\limits^{}_{} {\frac{x}{\sqrt{8x-10}}} \, dx =\frac{1}{\sqrt8}\int\limits^{}_{} {\frac{x}{\sqrt{x-1.25}}} \, dx\\\\=\frac{1}{\sqrt8} \int\limits^{}_{} {\frac{(t^2+1.25)2t}{t}} \, dt =\frac{2}{\sqrt8} \int\limits^{}_{} {(t^2+1.25)} \, dt =\frac{1}{\sqrt2}(\frac{t^3}{3}+1.25t)+C\\\\=\frac{1}{3\sqrt2}(x-1.25)^{\frac{3}{2}}+1.25(x-1.25)^{\frac{1}{2}}+C$