`int_(0)^(2//3)(dx)/(4 + 9x^(2))` equals

Options

A. `(pi)/(6)`
B. `(pi)/(12)`
C. `(pi)/(24)`
D. `(pi)/(4)`

Answer

To solve the integral

\[ \int_{0}^{\frac{2}{3}} \frac{dx}{4 + 9x^2} \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integrand to match a standard form. The expression \(4 + 9x^2\) can be factored to make it easier to integrate.

\[ 4 + 9x^2 = 4(1 + \frac{9}{4}x^2) \]

Thus, we can rewrite the integral as:

\[ \int_{0}^{\frac{2}{3}} \frac{dx}{4(1 + \frac{9}{4}x^2)} = \frac{1}{4} \int_{0}^{\frac{2}{3}} \frac{dx}{1 + \frac{9}{4}x^2} \]

### Step 2: Apply the Formula
We can use the formula for the integral of the form \(\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C\). Here, \(a^2 = \frac{4}{9}\) implies \(a = \frac{2}{3}\).

So, we rewrite our integral using this formula:

\[ \frac{1}{4} \cdot \frac{1}{\frac{2}{3}} \int_{0}^{\frac{2}{3}} \frac{dx}{1 + \left(\frac{3}{2}x\right)^2} \]

This simplifies to:

\[ \frac{3}{8} \int_{0}^{\frac{2}{3}} \frac{dx}{1 + \left(\frac{3}{2}x\right)^2} \]

### Step 3: Evaluate the Integral
Using the formula, we have:

\[ \int \frac{dx}{1 + \left(\frac{3}{2}x\right)^2} = \frac{2}{3} \tan^{-1}\left(\frac{3}{2}x\right) + C \]

Now, we need to evaluate this from \(0\) to \(\frac{2}{3}\):

\[ \frac{3}{8} \left[ \frac{2}{3} \tan^{-1}\left(\frac{3}{2} \cdot \frac{2}{3}\right) - \frac{2}{3} \tan^{-1}(0) \right] \]

Calculating the limits:

\[ \tan^{-1}(1) = \frac{\pi}{4} \quad \text{and} \quad \tan^{-1}(0) = 0 \]

So we have:

\[ \frac{3}{8} \left[ \frac{2}{3} \cdot \frac{\pi}{4} - 0 \right] = \frac{3}{8} \cdot \frac{2\pi}{12} = \frac{3\pi}{48} = \frac{\pi}{16} \]

### Final Answer
Thus, the value of the integral is:

\[ \frac{\pi}{16} \]