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The given integral is:

\[
\int_0^{\frac{\pi}{2}} \frac{\sin(x)}{1 + \cos(x) + \sin(x)} \, dx
\]

To solve this, let's go step by step.

### Step 1: Simplifying the denominator
We have the denominator \( 1 + \cos(x) + \sin(x) \). One way to approach this problem is by rewriting this expression using a trigonometric identity.

#### Try substitution
We could try using the substitution:
\[
u = \tan\left(\frac{x}{2}\right)
\]
With this substitution, trigonometric functions can be expressed in terms of \( u \), and the integral bounds also change accordingly.

However, let's first verify if there’s a simpler approach using symmetry or direct integration techniques.

I will now proceed with the computation.The solution to the given integral is:

\[
\int_0^{\frac{\pi}{2}} \frac{\sin(x)}{1 + \cos(x) + \sin(x)} \, dx = \frac{\pi}{4} - \frac{\ln(2)}{2}
\]

Would you like more details about the steps taken, or do you have any specific questions?

Here are some related questions you might find helpful:

1. How do trigonometric substitutions simplify integrals like this?
2. What are other common methods to solve integrals involving sine and cosine?
3. Can you explain how symmetry properties of functions help in integral evaluation?
4. How is the integral result involving \(\ln(2)\) derived in this case?
5. Are there any geometric interpretations of integrals involving trigonometric functions?

**Tip:** When dealing with integrals involving trigonometric functions, look out for symmetry or substitutions that simplify the relationship between sine and cosine.

Solve the integral: \( \int_0^{\frac{\pi}{2}} \frac{\sin(x)}{1 + \cos(x) + \sin(x)} \, dx \)

Learn how to solve the integral \( \int_0^{\frac{\pi}{2}} \frac{\sin(x)}{1 + \cos(x) + \sin(x)} \, dx \) using trigonometric substitution and integration techniques. This problem covers advanced calculus concepts and step-by-step solutions suitable for university-level students.

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