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!