The given problem is to evaluate the definite integral:

$\int_{1}^{4} 3x^2 \sqrt{x^3 + x} \, dx$

Solution:

1. Substitution: Let: $u = x^3 + x$ Then, the derivative is: $du = (3x^2 + 1) dx$

   From the integral, we see $3x^2 dx$ appears directly, but we adjust for $(+1)$: $dx = \frac{du}{3x^2 + 1}$

2. Adjust Limits: When $x = 1$: $u = 1^3 + 1 = 2$ When $x = 4$: $u = 4^3 + 4 = 68$

3. Substituted Integral: Replace in the integral: $\int_{1}^{4} 3x^2 \sqrt{x^3 + x} \, dx = \int_{2}^{68} \sqrt{u} \, \frac{du}{1} = \int_{2}^{68} u^{\frac{1}{2}} \, du$

4. Integrate $u^{\frac{1}{2}}$: $\int u^{\frac{1}{2}} \, du = \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} u^{\frac{3}{2}}$

5. Apply Limits: Substituting the limits: $\frac{2}{3} \left[ u^{\frac{3}{2}} \right]_{2}^{68}$ Compute: $\frac{2}{3} \left[ 68^{\frac{3}{2}} - 2^{\frac{3}{2}} \right]$ Calculate terms:

   • $68^{\frac{3}{2}} = \sqrt{68}^3 = 17.6^3 \approx 5456.576$
   • $2^{\frac{3}{2}} = \sqrt{2}^3 = 2.828^3 \approx 11.314$

   Subtract: $\frac{2}{3} \left( 5456.576 - 11.314 \right) = \frac{2}{3} \cdot 5445.262 \approx 3630.175$

Final Answer:

$\boxed{3630.175}$

Let me know if you'd like further clarification!

Expanding Questions:

1. How do substitutions simplify evaluating definite integrals?
2. What are other techniques to evaluate integrals involving square roots?
3. How can we verify the result of an integral using a numerical approach?
4. Can this integral be solved without substitution? If so, how?
5. What happens if the limits of integration are reversed?

Tip:

Always verify the adjusted limits after substitution, as they must match the new variable of integration.