To solve the integral

$\int (x\sqrt{x} + 3) \, dx,$

we break it down into manageable components.

Step 1: Simplify the Expression

We rewrite $x\sqrt{x}$ using the property of exponents:

$x\sqrt{x} = x^{3/2}.$

So, the integral becomes:

$\int (x^{3/2} + 3) \, dx.$

Step 2: Integrate Each Term

The integral can be separated into two parts:

$\int x^{3/2} \, dx + \int 3 \, dx.$

1. Integrate $x^{3/2}$: Use the power rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$).

   Here, $n = 3/2$, so:

   $\int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5}x^{5/2}.$

2. Integrate $3$: For constants, the integral is:

   $\int 3 \, dx = 3x.$

Step 3: Combine the Results

Adding the two parts together, we get:

$\int (x\sqrt{x} + 3) \, dx = \frac{2}{5}x^{5/2} + 3x + C,$

where $C$ is the constant of integration.

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Final Answer:

$\boxed{\frac{2}{5}x^{5/2} + 3x + C.}$

Would you like a step-by-step explanation of power rule integration or an example for a similar integral?

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5 Related Questions:

1. How do you handle integrals with fractional exponents?
2. What is the difference between definite and indefinite integrals?
3. How can you simplify radicals before integrating?
4. What happens if there is a coefficient before $x\sqrt{x}$?
5. How do you compute definite integrals for this type of problem?

Tip:

When integrating expressions with fractional exponents, always simplify the exponents and use the power rule systematically.