`int sqrt(9 -4x^(2)) dx`

Answer

To solve the integral \( \int \sqrt{9 - 4x^2} \, dx \), we can follow these steps:

### Step 1: Factor out the constant from the square root
We can rewrite the integral by factoring out the constant \(4\) from the expression under the square root:

\[ \int \sqrt{9 - 4x^2} \, dx = \int \sqrt{4 \left(\frac{9}{4} - x^2\right)} \, dx \]

### Step 2: Simplify the square root
Since \( \sqrt{4} = 2 \), we can take \(2\) out of the integral:

\[ = 2 \int \sqrt{\frac{9}{4} - x^2} \, dx \]

### Step 3: Substitute \( a \) and use the formula
Let \( a = \frac{3}{2} \). We can now rewrite the integral in terms of \( a \):

\[ = 2 \int \sqrt{a^2 - x^2} \, dx \]

We will use the formula for the integral \( \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C \).

### Step 4: Apply the formula
Substituting \( a = \frac{3}{2} \):

\[ = 2 \left( \frac{x}{2} \sqrt{\left(\frac{3}{2}\right)^2 - x^2} + \frac{\left(\frac{3}{2}\right)^2}{2} \sin^{-1}\left(\frac{2x}{3}\right) \right) + C \]

### Step 5: Simplify the expression
Now, we simplify the expression:

1. The first term becomes:
\[ 2 \cdot \frac{x}{2} \sqrt{\frac{9}{4} - x^2} = x \sqrt{\frac{9}{4} - x^2} \]

2. The second term becomes:
\[ 2 \cdot \frac{\frac{9}{4}}{2} \sin^{-1}\left(\frac{2x}{3}\right) = \frac{9}{4} \sin^{-1}\left(\frac{2x}{3}\right) \]

### Final Result
Putting it all together, we have:

\[ \int \sqrt{9 - 4x^2} \, dx = x \sqrt{9 - 4x^2} + \frac{9}{4} \sin^{-1}\left(\frac{2x}{3}\right) + C \]