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The given problem asks to evaluate the definite integral:

\[
\int_0^4 (3\sqrt{x} + 10x\sqrt{x}) \, dx
\]

Let's break down the expression and solve the integral:

1. **First term:** \( \int 3\sqrt{x} \, dx = \int 3x^{1/2} \, dx \)
   - Using the power rule for integration: 
   \[
   \int x^n \, dx = \frac{x^{n+1}}{n+1}
   \]
   So, 
   \[
   \int 3x^{1/2} \, dx = 3 \cdot \frac{x^{3/2}}{3/2} = 2x^{3/2}
   \]

2. **Second term:** \( \int 10x\sqrt{x} \, dx = \int 10x^{3/2} \, dx \)
   - Again, using the power rule for integration:
   \[
   \int 10x^{3/2} \, dx = 10 \cdot \frac{x^{5/2}}{5/2} = 4x^{5/2}
   \]

Now, combining the two results:
\[
\int_0^4 (3\sqrt{x} + 10x\sqrt{x}) \, dx = \left[ 2x^{3/2} + 4x^{5/2} \right]_0^4
\]

We evaluate at the upper and lower limits:

- At \(x = 4\):
  \[
  2(4^{3/2}) + 4(4^{5/2}) = 2(8) + 4(32) = 16 + 128 = 144
  \]

- At \(x = 0\), both terms are zero.

Thus, the value of the integral is 144.

The correct answer is **c. 144**.

---

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How do you apply the power rule to integrals involving square roots?
2. What is the result of integrating \( x^n \) for different values of \( n \)?
3. How does one approach definite integrals with polynomial terms?
4. How do you find the area under a curve using integration?
5. Can you explain the difference between definite and indefinite integrals?

**Tip:** Always double-check your limits of integration after calculating an indefinite integral, as plugging in values incorrectly can affect the final result.

Evaluate the integral \( \int_0^4 (3\sqrt{x} + 10x\sqrt{x}) \, dx \) and select the correct answer from the choices.

Learn how to evaluate the definite integral \( \int_0^4 (3\sqrt{x} + 10x\sqrt{x}) \, dx \) using the power rule of integration. This problem covers integral calculus concepts such as square roots and polynomial expressions. Suitable for advanced high school or early college students.

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