Euler substitution is a method for evaluating integrals of the form
where is a rational function of and . It is proved that these integrals can always be rationalized using one of three Euler substitutions.
The first substitution of Euler is used when . We substitute
and solve the resulting expression for . We have that and that the term is expressible rationally in .
In this substitution, either the positive sign or the negative sign can be chosen.
If , we take
We solve for similarly as above and find
Again, either the positive or the negative sign can be chosen.
If the polynomial has real roots and , we may choose . This yields
and as in the preceding cases, we can express the entire integrand rationally in .
For the integral , we can use the first substitution and set . Thus,
Accordingly, we obtain:
The cases give the formulas
For finding the value of
we find using the first substitution of Euler: . Squaring both sides of the equation gives us , from which the terms will cancel out. Solving for yields
From there, we find that the differentials and are related by
Hence,
In the integral
we can use the second substitution and set . Thus
and
Accordingly, we obtain:
To evaluate
we can use the third substitution and set . Thus
and
Next,
This is a rational function, which can be solved using partial fractions.
The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral , the substitution can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
The substitutions of Euler can be generalized to a larger class of functions. Consider integrals of the form
where and are rational functions of and . This integral can be transformed by the substitution into another integral
where and are now simply rational functions of . In principle, factorization and partial fraction decomposition can be employed to break the integral down into simple terms, which can be integrated analytically through use of the dilogarithm function.