오일러 치환
\(y^2=ax^2+bx+c\)를 \(t\)에 대한 유리함수로 매개화하는 것이 가장 중요한 아이디어이다.
\(y-y_0 = t(x-x_0)\) passing through a point \((x_0,y_0)\)
오일러치환
형태의 적분을 유리함수의 적분으로 바꾸는 변수 \(x=x(t)\) 치환
제1오일러치환
제2오일러치환
The third Euler substitution: If , then
제3오일러치환
The second Euler substitution: If the roots and of the quadratic polynomial are real, then
In the case when , that is, when (2) is a hyperbola, the first Euler substitution is obtained by taking \((x_0,y_0)\) as one of the points at infinity defined by the directions of the asymptotes of this hyperbola;
when the roots and of the quadratic polynomial \(ax^2+bx+c\) are real, the second Euler substitution is obtained by taking as \((x_0,y_0)\) one of the points or ;
finally, when , the third Euler substitution is obtained by taking as \((x_0,y_0)\) one of the points where the curve (2) intersects the ordinate axis, that is, one of the points .
http://www.integral-table.com/
\(\int \sqrt{x^2+1}\,dx\)
http://www.goiit.com/posts/list/integration-euler-s-substitution-354.htm
http://pauli.uni-muenster.de/~munsteg/arnold.html
multiply out. Since we can factor the polynomial and one root is 2, we can also use the 3. Euler substitution:
Euler substitutions Substitutions of the variable in an integral
where is a rational function of its arguments, that reduce (1) to the integral of a rational function. There are three types of such substitutions.
(Any combination of signs may be chosen on the right-hand side in each case.) All the Euler substitutions allow both the original variable of integration and to be expressed rationally in terms of the new variable .
The first two Euler substitutions permit the reduction of (1) to the integral of a rational function over any interval on which takes only real values.