Plotting an integral over a range of values

Question

I would like to evaluate the following integral

$$\int_{1}^{(x^{2}+1)/2x} dy \sqrt{y^{2}-1} \left(\frac{2x^{2}-6xy+3 + x^{2}y^{2}}{x^{2}\left(x-2y\right)^{2}}\right)$$

over a range of values of $x$ from $0$ to $2$ and plot the resulting integral.

Can you help me with this?

Answer 1

This integrates faster if we just integrate it as indefinite and then use FTC. Assuming the integral is proper, which I did not check.

ClearAll[x, y, k];
num = 2 x^2 - 6 x y + 3 + x^2 y^2;
den = x^2 (x - 2 y)^2;
int = Integrate[Sqrt[y^2 - 1] num/den, y]

upper = Limit[int, y -> (x^2 + 1)/(2 x)];
lower = Limit[int, y -> 1];
(upper - lower) // Simplify

f[x_] := Evaluate[upper - lower];

Plot[Chop@f[x], {x, 0, 2}, Frame -> True, 
 FrameLabel -> {{"integral", None}, {"x", "integral over x"}}, 
 GridLines -> Automatic, GridLinesStyle -> LightGray, 
 Exclusions -> None, BaseStyle -> 14, PlotStyle -> Red]