To solve the integral [tex]\(\int x^3 \sqrt{1 + x^2} \, dx\)[/tex], we can proceed by finding an antiderivative. Here's a detailed step-by-step solution:
1. Transformation to an Easier Form (if needed):
Since the integrand [tex]\( x^3 \sqrt{1 + x^2} \)[/tex] is not directly integrable through simple techniques, we consider possible substitutions or integrals rules. Based on known integration strategies, we can see if any common integral formulations apply here. In this case, the recognition of patterns and previously computed integrals lead us to the result.
2. Solve the Integral:
Given the integral in a more standard integral table or known result, the solution to [tex]\(\int x^3 \sqrt{1 + x^2} \, dx\)[/tex] is:
[tex]\[
\int x^3 \sqrt{1 + x^2} \, dx = \frac{x^4 \sqrt{x^2 + 1}}{5} + \frac{x^2 \sqrt{x^2 + 1}}{15} - \frac{2 \sqrt{x^2 + 1}}{15} + C
\][/tex]
3. Verification (Optional):
If needed, we could differentiate the antiderivative back to ensure it matches the original integrand, confirming the correctness.
Thus, the final solution to the integral [tex]\(\int x^3 \sqrt{1 + x^2} \, dx\)[/tex] is:
[tex]\[
\boxed{\frac{x^4 \sqrt{x^2 + 1}}{5} + \frac{x^2 \sqrt{x^2 + 1}}{15} - \frac{2 \sqrt{x^2 + 1}}{15} + C}
\][/tex]
Here, [tex]\(C\)[/tex] represents the constant of integration.
