`int 1/(3sqrt(x)) dx = ?`

Options

A. `3/2 x^(2/3) + C`
B. `3/(2x^(2/3)) + C`
C. `2/(3^(2/3)) + C`
D. `2/3x^(3/2) + C`

Answer

To solve the integral \( \int \frac{1}{3\sqrt{x}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{1}{3\sqrt{x}} \, dx \]
We can express \( \sqrt{x} \) as \( x^{1/2} \). Therefore, we can rewrite the integral as:
\[ \int \frac{1}{3x^{1/2}} \, dx \]

### Step 2: Factor Out the Constant
The constant \( \frac{1}{3} \) can be factored out of the integral:
\[ \frac{1}{3} \int x^{-1/2} \, dx \]

### Step 3: Apply the Power Rule for Integration
Now, we will apply the power rule for integration. The power rule states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
In our case, \( n = -\frac{1}{2} \). Therefore, we have:
\[ \int x^{-1/2} \, dx = \frac{x^{(-1/2)+1}}{(-1/2)+1} + C = \frac{x^{1/2}}{1/2} + C \]
This simplifies to:
\[ 2x^{1/2} + C \]

### Step 4: Combine the Results
Now we substitute back into our equation:
\[ \frac{1}{3} \left( 2x^{1/2} + C \right) = \frac{2}{3} x^{1/2} + \frac{C}{3} \]
Since \( C \) is an arbitrary constant, we can denote \( \frac{C}{3} \) as just \( C \). Thus, we get:
\[ \frac{2}{3} \sqrt{x} + C \]

### Final Answer
The final result of the integral is:
\[ \int \frac{1}{3\sqrt{x}} \, dx = \frac{2}{3} \sqrt{x} + C \]

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