3
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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]

Mathematica graphics

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

Mathematica graphics

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]

Mathematica graphics

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.