Intereting Posts

Prove that $(A-B) \cap (A-C) = A \cap (B \cup C)^c$ for any three sets A, B, C.
Replacing in equation introduces more solutions
Hilbert class field of a quadratic field whose class number is 3
Linear algebra problem from dummite & foote
Intuition explanation of taylor expansion?
Diophantine number has full measure but is meager
Unexpected examples of natural logarithm
Tangent space to circle
Calculating prime numbers
Does such localization of integral extension preserve inclusion?
Classical texts that should not be missing from any shelf
Full-rank condition for product of two matrices
Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?
Additive functional inequality
Are any of these notions of “k-space” equivalent if $X$ is not assumed weakly Hausdorff?

I believe there is a way to geometrically interpret integrating $R(x,\sqrt{Ax^2+Bx+C})$, as a means to motivate the Euler substitutions, in terms of

“…expressing the coordinates of a point upon a conic $y^2 = Ax^2+Bx+C$ by means of

rational functions of a parameter. It can be seen geometrically that this is possible,. For, if a secant $y – b = t(x – a)$ be drawn through any point $(a,b)$ on the conic, the coordinates of the second point of intersection of the secant with the conic are given by equations of the first degree, and are therefore rational functions of $t$”.Goursat – Page 215

- Evaluation of $\int\frac{dx}{x+ \sqrt{x^2-x+1}}$
- How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$
- What is the correct integral of $\frac{1}{x}$?
- Evaluate $\int\frac{1}{1+x^6} \,dx$
- How to evaluate $\int 1/(1+x^{2n})\,dx$ for an arbitrary positive integer $n$?
- What is the integral of $e^x \tan(x)$?

Can anyone parse this for me? I don’t really see how this shows the second point of intersection is a rational function of the first degree.

This interpretation allows one to geometrically see that if the quadratic has imaginary roots, $A > 0$ and so this immediately implies the quadratic is a hyperbola. Therefore, geometrically, we see our ‘Euler substitution’ simply must be $y = \sqrt{A}x + t$, a straight line parallel to an asymptote of the hyperbola, and because it’s an asymptote it cuts the hyperbola at the point

$$x = \frac{C-t^2}{2\sqrt{A}t-2B}, y = t+\sqrt{A}\frac{C-t^2}{2\sqrt{A}-2B}.$$

Similarly, if $A < 0$ the conic simply must be an ellipse, otherwise the quadratic is negative, and so geometrically we see we should substitute $y=t(x-a)$, the moving secant cutting the conic.

I believe this is an answer to this question, however I’d like somebody to parse all this out in a clearer form for us all to read, i.e. to make some of it more concrete and mathematical.

If one is masochistic enough, they could relate it to this interesting post also, but that’s not necessary, thank you

- Evaluating the primitive $\int \frac{\mathrm dx}{e^{2x} + e^x + 1} $
- Putnam Exam Integral
- Integral of Thomae's function
- Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$
- how to find $\int_{0}^{1}h_n(x)dx?$
- Hydrostatic pressure on an equilateral triangle
- On the integral $\int_0^1\frac{dx}{\sqrtx\ \sqrt{1-x}\ \sqrt{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$
- Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$
- Integrating exponential of exponential function
- A Bound for the Error of the Numerical Approximation of a the Integral of a Continuous Function

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- Multivariable Calculus for GRE
- Is gcd the right adjoint of something?
- Formula for calculating residue at a simple pole.
- Finding value of an expression
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- formula for calculating determinant of the block matrix
- Using Carmichael function in RSA.
- How can I find $\lim_{n\to \infty} a_n$
- Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?
- General and Simple Math Problem.
- Does using an ellipse as a template still produce an ellipse?
- Homework: No field extension is “degree 4 away from an algebraic closure”
- What does it mean for something to be true but not provable in peano arithmetic?