Euler substitutions are used for solving integrals of the form

$$\int R(x, \sqrt{ax^2+bx+c}) dx \tag{1} $$ where $R$ is a rational two-argument function.

There is plenty of information about them on the web so this post will be just very short.

1) The first Euler substitution is defined by

$\sqrt{ax^2+bx+c} = \sqrt{a} \cdot x + t \tag{2}$

It is used when $a \gt 0$

2) The second Euler substitution is defined by

$\sqrt{ax^2+bx+c} = x \cdot t + \sqrt{c} \tag{3}$

It is used when $c \gt 0$

3) The third Euler substitution is used when the quadratic polynomial

$ax^2+bx+c$ has 2 distinct real roots $\alpha$ and $\beta$.

It is defined by the equality

$\sqrt{a(x-\alpha)(x-\beta)} = t \cdot (x-\alpha) \tag{4}$

The equality $(2), (3),$ or $(4)$ is then solved for $x$, and $x$ is replaced in $(1)$ with the respective resulting expression/function of $t$. This allows us to transform the integral $(1)$ into an integral from a rational function of $t$.

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