Integrating without using Euler substitution.

Question

$\int\frac{1}{\sqrt{(x^2-2x+5)}}dx$

I wonder if there is a way to solve this without using Euler substitution.

Answer 1

$x^2-2x+5=(x-1)^2+4$

Replace $x-1=2\sinh(y)$. Recall one of the definitions of $\sinh$$$\sinh(y)=\frac{e^{y}-e^{-y}}{2}\ \text{ and }\ \cosh(y)=\frac{e^{y}+e^{-y}}{2}$$

Use that the hyperbolic functions parametrize the hyperbola $$\cosh^2(y)-\sinh^2(y)=1$$

and that the derivative of $\sinh(y)$ is $\cosh(y)$.

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Trigonometric and hyperbolic substitutions for this type of problems are not real friends. As soon as the problem changes a little they will abandon you. Euler's will always work for you for integrals of rational functions of quadratic radicals.