Question

what is the integration of 1/sqrt of [(x-1)(x-2)(x-3)] dx

Answer 1

Answer on Question #51629 – Math – Integral Calculus

What is the integration of $\frac{1}{\sqrt{(x - 1)(x - 2)(x - 3)}}$

Solution

Let

$$I(x) = \int \frac{1}{\sqrt{(x - 1)(x - 2)(x - 3)}} \, dx.$$

This integral cannot be expressed in elementary functions.

We use Wolfram Mathematica online integrator and find

$$I(x) = \frac{\left(2i\sqrt{\left(\frac{1}{x - 3} + 1\right)} \sqrt{\left(\frac{2}{x - 3} + 1\right)} (x - 3)^{\frac{3}{2}} \cdot F\left(i \sinh^{-1}\left(\frac{1}{\sqrt{x - 3}}\right) \mid 2\right)\right)}{\sqrt{(x - 1)(x - 2)(x - 3)}},$$

where $i$ is imaginary unit, $\sinh^{-1}(x)$ is inverse hyperbolic sine, $F(x|m)$ is elliptic integral of the first kind.

We can simplify it to

$$I(x) = \frac{\left(2i\sqrt{\left(\frac{x - 2}{x - 3}\right)} \sqrt{\left(\frac{x - 1}{x - 3}\right)} (x - 3)^{\frac{3}{2}} \cdot F\left(i \sinh^{-1}\left(\frac{1}{\sqrt{x - 3}}\right) \mid 2\right)\right)}{\sqrt{(x - 1)(x - 2)(x - 3)}} = 2i \cdot F\left(i \sinh^{-1}\left(\frac{1}{\sqrt{x - 3}}\right) \mid 2\right).$$

Answer: $2i \cdot F\left(i \sinh^{-1}\left(\frac{1}{\sqrt{x - 3}}\right) \mid 2\right)$.

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