Euler substitution is a method for evaluating integrals of the form:
where is a rational function of and . In such cases the integrand can be changed to a rational function of a new variable by using the following substitutions of Euler:
The first substitution of Euler a > 0
If a > 0 we may write When we take with the minus sign, then from which we get the expression thus also is expressible rationally via . We have .
The second substitution of Euler. c > 0
If c > 0 we take With the minus sign we obtain, similarly as above,
The third substitution of Euler
If the polynomial has the real zeros and , we may chose
This gives the expression
As in the preceding cases, we can express and rationally via .
|This section requires expansion. (March 2012)|
In the integral we can use the first substitution:, then and thus
Accordingly we obtain:
Especially the cases , give the formulas
- N. Piskunov, Diferentsiaal- ja integraalarvutus körgematele tehnilistele öppeasutustele. Viies, taiendatud trukk. Kirjastus Valgus, Tallinn (1965). Note: Euler substitutions can be found in most Russian calculus textbooks.