To solve the integral:

$\int 3\sqrt{x} \, dx$

Step 1: Rewrite the integrand

We can express the square root as a fractional exponent: $\sqrt{x} = x^{1/2}$ Thus, the integral becomes: $\int 3x^{1/2} \, dx$

Step 2: Apply the power rule

The power rule for integration states that: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where n \neq -1)} In this case, $n = \frac{1}{2}$. Applying the rule:

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

Step 3: Simplify

To simplify the fraction: $3 \cdot \frac{x^{3/2}}{3/2} = 3 \cdot \frac{2}{3} \cdot x^{3/2} = 2x^{3/2}$

Final answer:

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

Would you like further explanation or additional steps?

Here are 5 related questions:

1. How do you solve integrals with negative exponents?
2. What is the integral of $\frac{1}{x}$?
3. How do you handle integrals with trigonometric functions?
4. How do definite integrals differ from indefinite integrals?
5. Can you explain the substitution method for integrals?

Tip: When dealing with powers, always remember to adjust the exponent and divide by the new exponent when integrating!