Euler substitution

Euler substitution is a method for evaluating integrals of the form:

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

where $R$ is a rational function of $x$ and $\sqrt{ax^2+bx+c}$. In such cases the integrand can be changed to a rational function of a new variable $t$ by using the following substitutions of Euler:

The first substitution of Euler a > 0

If a > 0 we may write $\sqrt{ax^2+bx+c} \;=\; \pm x\sqrt{a}+t.$ When we take $\sqrt{a}$ with the minus sign, then $ax^2+bx+c \;=\; ax^2-2xt\sqrt{a}+t^2$ from which we get the expression $x = \frac{t^2-c}{b+2t\sqrt{a}}$ thus also $dx$ is expressible rationally via $t$ . We have $\sqrt{ax^2+bx+c} \;=\; -x\sqrt{a}+t \;=\; \frac{c-t^2}{b+2t\sqrt{a}}\sqrt{a}+t$.

The second substitution of Euler. c > 0

If c > 0 we take $\begin{align} \sqrt{ax^2+bx+c} \;=\; xt\pm\sqrt{c}. \end{align}$ With the minus sign we obtain, similarly as above, $x \;=\; \frac{2t\sqrt{c}+b}{t^2-a}.$

The third substitution of Euler

If the polynomial $ax^2 + bx + c$ has the real zeros $\alpha$ and $\beta$, we may chose

$\sqrt{ax^2\!+\!bx\!+\!c} \;=\; (x\!-\!\alpha)t$

Now

$ax^2\!+\!bx\!+\!c \;=\; a(x\!-\!\alpha)(x\!-\!\beta) \;=\; (x\!-\!\alpha)^2t^2$

$a(x\!-\!\beta) = (x\!-\!\alpha)t^2$

This gives the expression

$x = \frac{a\beta-\alpha t^2}{a-t^2}$

As in the preceding cases, we can express $dx$ and $\sqrt{ax^2\!+\!bx\!+\!c}$ rationally via $t$.

Examples

In the integral $\int\ \frac{dx}{\sqrt{x^2+c}}$ we can use the first substitution:$\sqrt{x^2+c} = -x+t$, then $x^2+c = x^2-2xt+t^2$ and thus

$x = \frac{t^2-c}{2t} \quad\quad dx = \frac{t^2+c}{2t^2} dt$

$\sqrt{x^2+c} = -\frac{t^2-c}{2t}+t = \frac{t^2+c}{2t}$

Accordingly we obtain:

$\int \frac{dx}{\sqrt{x^2+c}} = \int \frac{\frac{t^2+c}{2t^2}dt}{\frac{t^2+c}{2t}}$

$= \int\!\frac{dt}{t} = \ln|t|+C = \ln|x+\sqrt{x^2+c}|+C$

Especially the cases $c = \pm 1$, give the formulas

$\int \frac{dx}{\sqrt{x^2+1}} = \mbox{arcsinh}(x) + C$

$\int \frac{dx}{\sqrt{x^2-1}} = \mbox{arccosh}(x) + C \quad (x > 1)$

References

This article incorporates material from Eulers Substitutions For Integration on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.