To solve the integral:

$I = \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \sin x}$

by writing $\sin x$ in terms of $\tan \left(\frac{x}{2}\right)$, we use the following trigonometric identities based on the Weierstrass substitution:

$\sin x = \frac{2 \tan \left( \frac{x}{2} \right)}{1 + \tan^2 \left( \frac{x}{2} \right)}$ $dx = \frac{2}{1 + \tan^2 \left( \frac{x}{2} \right)} d\left(\tan \left( \frac{x}{2} \right)\right)$

Now, follow these steps:

Step 1: Substitution

Let $t = \tan \left( \frac{x}{2} \right)$. Thus, we have: $\sin x = \frac{2t}{1 + t^2}, \quad dx = \frac{2 \, dt}{1 + t^2}$ Also, when $x = \frac{\pi}{4}$, $t = 1$, and when $x = \frac{3\pi}{4}$, $t = 1$. So the limits of integration will be from $1$ to $1$.

Step 2: Substitute into the integral

Substitute these into the integral $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \sin x}$:

$I = \int_1^1 \frac{\frac{2 \, dt}{1 + t^2}}{1 + \frac{2t}{1 + t^2}} = \int_1^1 \frac{2 \, dt}{(1 + t^2) + 2t}$

Simplify the denominator: $(1 + t^2) + 2t = (1 + t)^2$ Thus, the integral becomes: $I = \int_1^1 \frac{2 \, dt}{(1 + t)^2}$

Step 3: Perform the integration

Now, the integral simplifies to: $I = \int_1^1 \frac{2 \, dt}{(1 + t)^2}$

Since the limits of integration are from 1 to 1, the result of the integral is: $I = 0$

Final Answer:

The value of the integral is $\boxed{0}$.

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Would you like more details on any part of this solution or have any further questions?

Here are 5 related questions for practice:

1. How do you derive the Weierstrass substitution?
2. What is the integral of $\frac{1}{1 + \sin x}$ using direct trigonometric identities without substitution?
3. Solve $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$.
4. What is the relationship between $\sin x$ and $\tan \left( \frac{x}{2} \right)$ in terms of their geometric interpretation?
5. How do you handle integrals where the limits of integration are the same?

Tip: Always check the limits of integration before simplifying, as this can sometimes make the problem much easier!