Euler substitution

Euler substitution is a method for evaluating integrals of the form

$${\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx,}$$

where ${\displaystyle R}$ is a rational function of ${\displaystyle x}$ and ${\textstyle {\sqrt {ax^{2}+bx+c}}}$. It is proved that these integrals can always be rationalized using one of three Euler substitutions.

The first substitution of Euler is used when ${\displaystyle a>0}$. We substitute $${\displaystyle {\sqrt {ax^{2}+bx+c}}=\pm x{\sqrt {a}}+t}$$ and solve the resulting expression for ${\displaystyle x}$. We have that ${\displaystyle x={\frac {c-t^{2}}{\pm 2t{\sqrt {a}}-b}}}$ and that the ${\displaystyle dx}$ term is expressible rationally in ${\displaystyle t}$.

In this substitution, either the positive sign or the negative sign can be chosen.

If ${\displaystyle c>0}$, we take $${\displaystyle {\sqrt {ax^{2}+bx+c}}=xt\pm {\sqrt {c}}.}$$ We solve for ${\displaystyle x}$ similarly as above and find $${\displaystyle x={\frac {\pm 2t{\sqrt {c}}-b}{a-t^{2}}}.}$$

Again, either the positive or the negative sign can be chosen.

If the polynomial ${\displaystyle ax^{2}+bx+c}$ has real roots ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, we may choose ${\textstyle {\sqrt {ax^{2}+bx+c}}={\sqrt {a(x-\alpha )(x-\beta )}}=(x-\alpha )t}$. This yields ${\displaystyle x={\frac {a\beta -\alpha t^{2}}{a-t^{2}}},}$ and as in the preceding cases, we can express the entire integrand rationally in ${\displaystyle t}$.