`int 3sqrt(x) dx = ?`

Options

A. `3/4x^(3/4) + C`
B. `4/3x^(3/4) + C`
C. `3/4 x^(4/3) +C`
D. `4/3 x^(4/3) +C`

Answer

To solve the integral \( \int 3\sqrt{x} \, dx \), we will follow these steps:

### Step 1: Rewrite the integrand
We start by rewriting \( \sqrt{x} \) in terms of exponents:
\[ \sqrt{x} = x^{1/2} \]
Thus, we can rewrite the integral as:
\[ \int 3\sqrt{x} \, dx = \int 3x^{1/2} \, dx \]

### Step 2: Factor out the constant
Next, we can factor out the constant \( 3 \) from the integral:
\[ \int 3x^{1/2} \, dx = 3 \int x^{1/2} \, dx \]

### Step 3: Apply the power rule of integration
Now we apply the power rule of integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In our case, \( n = \frac{1}{2} \):
\[ \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C = \frac{x^{3/2}}{3/2} + C \]
This simplifies to:
\[ \int x^{1/2} \, dx = \frac{2}{3} x^{3/2} + C \]

### Step 4: Substitute back into the integral
Now we substitute this result back into our expression:
\[ 3 \int x^{1/2} \, dx = 3 \left( \frac{2}{3} x^{3/2} + C \right) \]

### Step 5: Simplify the expression
Distributing the \( 3 \):
\[ = 2x^{3/2} + 3C \]
Since \( 3C \) is still a constant, we can denote it as \( C \):
\[ = 2x^{3/2} + C \]

### Final Answer
Thus, the final answer is:
\[ \int 3\sqrt{x} \, dx = 2x^{3/2} + C \]

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