`int(3x sqrtx +4 sqrtx+5) dx=`

Options

A. `6/5 x^(5/2) -8/3 x^(3/2) +5x+c`
B. `6/5 x^(5/2)+8/3 x^(3/2)+5x+c`
C. `6/5 x^(5/2)+4/3x^(3/2)+5x+c`
D. `4/5 x^(5/2)+8/3 x^3/2 +5x+c`

Answer

To solve the integral \( \int (3x \sqrt{x} + 4 \sqrt{x} + 5) \, dx \), we will break it down step by step.

### Step 1: Rewrite the integrand
First, we can rewrite the terms involving square roots in terms of exponents:
- \( \sqrt{x} = x^{1/2} \)
- Therefore, \( 3x \sqrt{x} = 3x^{1} \cdot x^{1/2} = 3x^{3/2} \)
- And \( 4 \sqrt{x} = 4x^{1/2} \)

Now we can rewrite the integral:
\[ \int (3x^{3/2} + 4x^{1/2} + 5) \, dx \]

### Step 2: Separate the integral
Using the property of integrals, we can separate the integral into three parts:
\[ \int 3x^{3/2} \, dx + \int 4x^{1/2} \, dx + \int 5 \, dx \]

### Step 3: Integrate each term
Now we will integrate each term separately using the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

1. For \( \int 3x^{3/2} \, dx \):
\[ = 3 \cdot \frac{x^{3/2 + 1}}{3/2 + 1} = 3 \cdot \frac{x^{5/2}}{5/2} = 3 \cdot \frac{2}{5} x^{5/2} = \frac{6}{5} x^{5/2} \]

2. For \( \int 4x^{1/2} \, dx \):
\[ = 4 \cdot \frac{x^{1/2 + 1}}{1/2 + 1} = 4 \cdot \frac{x^{3/2}}{3/2} = 4 \cdot \frac{2}{3} x^{3/2} = \frac{8}{3} x^{3/2} \]

3. For \( \int 5 \, dx \):
\[ = 5x \]

### Step 4: Combine the results
Now we combine all the integrated terms:
\[ \int (3x \sqrt{x} + 4 \sqrt{x} + 5) \, dx = \frac{6}{5} x^{5/2} + \frac{8}{3} x^{3/2} + 5x + C \]

### Final Answer
Thus, the final answer is:
\[ \int (3x \sqrt{x} + 4 \sqrt{x} + 5) \, dx = \frac{6}{5} x^{5/2} + \frac{8}{3} x^{3/2} + 5x + C \]