SOLVE     INTEGRAL dx/1+sinx    FROM pi/4 TO 3pi/4                               BY WRITING SINX IN TERMS OF TAN

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 \( \boxed{0} \).

---

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!

SOLVE INTEGRAL dx/1+sinx FROM pi/4 TO 3pi/4 BY WRITING SINX IN TERMS OF TAN

Learn how to solve the integral of 1/(1 + sin x) from pi/4 to 3pi/4 by expressing sin x in terms of tan(x/2) using the Weierstrass substitution. This detailed solution covers trigonometric identities, substitution methods, and step-by-step integration techniques suitable for undergraduate calculus students.

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