To solve the integral [tex]\( \int x^2 \cdot\left(x^2-1\right)^{-\frac{3}{2}} \, dx \)[/tex], let's proceed with the solution step-by-step:
1. Identify the integrand:
[tex]\[
x^2 \cdot (x^2 - 1)^{-\frac{3}{2}}
\][/tex]
2. Simplify the integrand, if possible:
- In this case, there isn't a straightforward simplification that makes the integral easier using basic algebraic techniques, but it helps to recognize that the integrand involves a composition of polynomial and power functions.
3. Consider an appropriate substitution:
- For integrals involving expressions such as [tex]\(x^2 - 1\)[/tex], it's often effective to use a trigonometric or hyperbolic substitution, although it's also possible to work with direct antiderivatives that involve inverse trigonometric or hyperbolic functions.
4. Integral boundaries and types:
- The integral can be expressed in terms of functions that involve arcsin (inverse sine) or arcosh (inverse hyperbolic cosine) depending on the domain of [tex]\(x\)[/tex].
The resulting integral is a piecewise function due to the nature of [tex]\(x^2 - 1\)[/tex]. The integral solution can be written in different forms depending on the domain of [tex]\(x\)[/tex]. Here, the solution takes into account the absolute value of [tex]\(x^2\)[/tex]:
5. Integral solution:
[tex]\[
\int x^2 \cdot (x^2 - 1)^{-\frac{3}{2}} \, dx = \text{Piecewise}\left( \left(-0.564189583547756 \cdot \sqrt{\pi} \cdot \frac{x}{\sqrt{x^2 - 1}} + 0.564189583547756 \cdot \sqrt{\pi} \cdot \text{acosh}(x), \, \lvert x^2 \rvert > 1 \right), \, \left(0.564189583547756 \cdot i \cdot \sqrt{\pi} \cdot \frac{x}{\sqrt{1 - x^2}} - 0.564189583547756 \cdot i \cdot \sqrt{\pi} \cdot \text{asin}(x), \, \text{True}\right) \right)
\][/tex]
These pieces break down based on the condition:
- When [tex]\(|x| > 1\)[/tex], which means [tex]\(x^2 > 1\)[/tex]:
[tex]\[
-0.564189583547756 \cdot \sqrt{\pi} \cdot \frac{x}{\sqrt{x^2 - 1}} + 0.564189583547756 \cdot \sqrt{\pi} \cdot \text{acosh}(x)
\][/tex]
- Otherwise, when [tex]\(|x| \le 1\)[/tex]:
[tex]\[
0.564189583547756 \cdot i \cdot \sqrt{\pi} \cdot \frac{x}{\sqrt{1 - x^2}} - 0.564189583547756 \cdot i \cdot \sqrt{\pi} \cdot \text{asin}(x)
\][/tex]
This piecewise solution correctly handles both real and complex domains based on the value of [tex]\(x^2\)[/tex] relative to 1.
6. Simplified Result:
For clarity, the integral solution can also be written more concisely with typical notation conventions:
[tex]\[
\int x^2 \cdot (x^2 - 1)^{-\frac{3}{2}} \, dx = \text{Piecewise}\left(\left(-\frac{\sqrt{\pi}}{2} \cdot \frac{x}{\sqrt{x^2 - 1}} + \frac{\sqrt{\pi}}{2} \cdot \text{acosh}(x), \, (x > 1) \, \text{or} \, (x < -1)\right), \, \left(\frac{i \sqrt{\pi}}{2} \cdot \frac{x - \sqrt{1 - x^2} \cdot \text{asin}(x)}{\sqrt{1 - x^2}}, \, \text{otherwise}\right)\right)
\][/tex]
This solution accurately conveys the integral results with respect to the domains of the variable [tex]\(x\)[/tex].
