Certainly! Let's solve the integral step-by-step:
We need to find the antiderivative of the integral [tex]\(\int \frac{x^3}{\sqrt{x^2 - 2}} \, dx\)[/tex].
To solve this, we will break it down in detail:
1. Identify the integrand: The function to be integrated is [tex]\(\frac{x^3}{\sqrt{x^2 - 2}}\)[/tex].
2. Simplify the integrand: Notice how the numerator [tex]\(x^3\)[/tex] can be related to the differential of the denominator's inner function. Here, the inner function of the square root is [tex]\(x^2 - 2\)[/tex], and its derivative is [tex]\(2x\)[/tex]. This suggests that we might use a substitution method to simplify the integral.
3. Substitute: We let [tex]\(u = x^2 - 2\)[/tex], hence [tex]\(du = 2x \, dx\)[/tex] or [tex]\(dx = \frac{du}{2x}\)[/tex].
4. Express the integral in terms of [tex]\(u\)[/tex]: Substitute [tex]\(u\)[/tex] and [tex]\(dx\)[/tex] in the integral. However, this can become intricate. Another approach is to recognize a pattern or find parts that resemble the derivative of the assumed form in the back of our mind.
5. Observation: Notice that [tex]\(x^2 - 2\)[/tex] appears under the square root. Let's think about differentiating a function like [tex]\((x^2 - 2)^{3/2}\)[/tex] which will give us terms related to both [tex]\(x^3\)[/tex] and [tex]\(\sqrt{x^2 - 2}\)[/tex].
6. Integrate: Observing this, we can verify:
[tex]\[
\int \frac{x^3}{\sqrt{x^2 - 2}} \, dx = x^2 \frac{1}{3} \sqrt{x^2 - 2} + \frac{4}{3} \sqrt{x^2 - 2}
\][/tex]
7. Result:
Thus, the antiderivative of [tex]\(\int \frac{x^3}{\sqrt{x^2 - 2}} \, dx\)[/tex] is:
[tex]\[
\frac{x^2 \sqrt{x^2 - 2}}{3} + \frac{4 \sqrt{x^2 - 2}}{3} + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
In conclusion:
[tex]\[
\boxed{\frac{x^2 \sqrt{x^2 - 2}}{3} + \frac{4 \sqrt{x^2 - 2}}{3} + C}
\][/tex]
is the antiderivative of the given integral.
