The integral presented in the image is:

$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{dx}{\sin(2x)}$

To solve this, we will follow these steps:

Step 1: Use the identity for $\sin(2x)$

Recall that:

$\sin(2x) = 2 \sin(x) \cos(x)$

So, the integral becomes:

$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{dx}{2 \sin(x) \cos(x)}$

Step 2: Simplify the integral

We can express $\frac{1}{\sin(2x)}$ as:

$\frac{1}{\sin(2x)} = \frac{1}{2 \sin(x) \cos(x)} = \frac{1}{2} \left( \frac{1}{\sin(x) \cos(x)} \right)$

Now, observe that:

$\frac{1}{\sin(x) \cos(x)} = \frac{2}{\sin(2x)}$

Thus, the integral becomes:

$\frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{dx}{\sin(2x)}$

Step 3: Evaluate the integral

The integral of $\frac{1}{\sin(2x)}$ is the logarithmic function of $\tan(x)$, giving us:

$\int \frac{dx}{\sin(2x)} = -\frac{1}{2} \ln \left| \cot(x) \right|$

Evaluating this from $\frac{\pi}{4}$ to $\frac{\pi}{3}$:

$-\frac{1}{2} \left( \ln \left| \cot \left( \frac{\pi}{3} \right) \right| - \ln \left| \cot \left( \frac{\pi}{4} \right) \right| \right)$

Now, calculating these values:

• $\cot \left( \frac{\pi}{3} \right) = \frac{1}{\sqrt{3}}$
• $\cot \left( \frac{\pi}{4} \right) = 1$

So, we have:

$-\frac{1}{2} \left( \ln \left( \frac{1}{\sqrt{3}} \right) - \ln(1) \right) = -\frac{1}{2} \left( -\frac{1}{2} \ln(3) \right)$

Thus, the final value is:

$\frac{1}{4} \ln(3)$

Therefore, the value of the definite integral is:

$\boxed{\frac{1}{4} \ln(3)}$

Would you like further details or clarifications on any steps?

Here are 5 related questions to explore:

1. How do you compute integrals involving trigonometric identities?
2. Can integrals of other trigonometric functions be solved similarly?
3. How does the substitution method work in integral calculus?
4. What is the geometric interpretation of an integral?
5. How do definite integrals differ from indefinite integrals?

Tip: When dealing with trigonometric integrals, it's helpful to use known identities to simplify the expressions first before integrating.