The value of `int_((pi)/(3))^((pi)/(2))(2+3sin x)/(sin x(1+cos x))dx` is equal to

Options

A. `(7)/(2)-sqrt(3)-log_(e)sqrt(3)`
B. `(10)/(3)-sqrt(3)-log_(e)sqrt(3)`
C. `(10)/(3)-sqrt(3)+log_(e)sqrt(3)`
D. `-2+3sqrt(3)+log_(e)sqrt(3)`

Answer

To solve the integral

\[ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2 + 3 \sin x}{\sin x(1 + \cos x)} \, dx, \]

we can break it down into two separate integrals:

\[ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2}{\sin x(1 + \cos x)} \, dx + \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{3 \sin x}{\sin x(1 + \cos x)} \, dx. \]

This simplifies to:

\[ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2}{\sin x(1 + \cos x)} \, dx + 3 \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1 + \cos x} \, dx. \]

### Step 1: Evaluate the first integral

We can rewrite the first integral:

\[ \int \frac{2}{\sin x(1 + \cos x)} \, dx. \]

Using the identity \(1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right)\) and \(\sin x = 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right)\), we can rewrite it as:

\[ \int \frac{2}{2 \sin\left(\frac{x}{2}\right) \cos^2\left(\frac{x}{2}\right)} \, dx = \int \frac{1}{\sin\left(\frac{x}{2}\right) \cos^2\left(\frac{x}{2}\right)} \, dx. \]

### Step 2: Substitute \(t = \tan\left(\frac{x}{2}\right)\)

Using the substitution \(t = \tan\left(\frac{x}{2}\right)\), we have:

\[ dx = \frac{2}{1 + t^2} \, dt. \]

The limits change as follows:
- When \(x = \frac{\pi}{3}\), \(t = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\).
- When \(x = \frac{\pi}{2}\), \(t = \tan\left(\frac{\pi}{4}\right) = 1\).

Thus, the integral becomes:

\[ \int_{\frac{1}{\sqrt{3}}}^{1} \frac{2}{t(1 + t^2)} \cdot \frac{2}{1 + t^2} \, dt = 2 \int_{\frac{1}{\sqrt{3}}}^{1} \frac{1}{t(1 + t^2)} \, dt. \]

### Step 3: Evaluate the second integral

Now, we evaluate:

\[ 3 \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1 + \cos x} \, dx. \]

Using the same substitution \(t = \tan\left(\frac{x}{2}\right)\), we can evaluate this integral similarly.

### Step 4: Combine results

After evaluating both integrals, we combine the results to find the final answer.

### Final Answer

The value of the original integral is:

\[ \frac{1}{2} \ln(3) + \tan\left(\frac{\pi}{3}\right) - \sqrt{3}. \]