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The given problem is:

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

Let's solve it step by step.

### Step 1: Simplify the integrand
We can rewrite the terms involving square roots:
- \( \sqrt{x} = x^{1/2} \)
- \( 10x\sqrt{x} = 10x \cdot x^{1/2} = 10x^{3/2} \)

Thus, the integral becomes:

\[
\int_0^4 \left(3x^{1/2} + 10x^{3/2}\right)dx
\]

### Step 2: Integrate each term
We now integrate the terms individually:
- For \( 3x^{1/2} \), the integral is \( 3 \cdot \frac{2}{3} x^{3/2} = 2x^{3/2} \).
- For \( 10x^{3/2} \), the integral is \( 10 \cdot \frac{2}{5} x^{5/2} = 4x^{5/2} \).

Thus, the indefinite integral becomes:

\[
2x^{3/2} + 4x^{5/2}
\]

### Step 3: Evaluate the definite integral
Now, we evaluate the definite integral from 0 to 4:

\[
\left[ 2x^{3/2} + 4x^{5/2} \right]_0^4
\]

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

Thus, the value at \( x = 4 \) is:

\[
16 + 128 = 144
\]

#### At \( x = 0 \):
Both terms vanish at \( x = 0 \), since they contain powers of \( x \).

Thus, the definite integral is:

\[
144 - 0 = 144
\]

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

---

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore:
1. How do you integrate functions involving square roots of \( x \)?
2. What is the general formula for integrating powers of \( x \)?
3. How do definite integrals differ from indefinite integrals?
4. Can integrals involving radicals always be simplified using exponents?
5. What other techniques exist for solving more complex integrals?

**Tip:** When dealing with powers of \( x \), converting square roots and radicals into fractional exponents simplifies the integration process!

Evaluate the integral: ∫[0 to 4] (3√x + 10x√x) dx

Solve the definite integral ∫[0 to 4] (3√x + 10x√x) dx using the power rule for integration. This solution involves simplifying radicals and applying integration techniques for powers of x.

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