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.