SOLVE     INTEGRAL dx/1+sinx    FROM pi/4 TO 3pi/4                               BY WRITING SINX IN TERMS OF TAN  BUT YOU SHOULD NOT CHANGE LIMITS THAT IT PUT VALUES IN THE FINAL RESULT AFTER RESUBSTITUTING

We are asked to solve the integral:

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

using the substitution \( \sin x \) in terms of \( \tan\), but without changing the limits, which we will reapply after resubstitution. Here's how to approach this:

### Step 1: Substitution
We use the Weierstrass substitution, which is based on the half-angle identity:
\[
\tan\left(\frac{x}{2}\right) = t
\]
Thus, the following relations hold:
\[
\sin x = \frac{2t}{1 + t^2}, \quad dx = \frac{2\, dt}{1 + t^2}
\]

### Step 2: Rewrite the integrand
Using these, the integral becomes:

\[
I = \int \frac{dx}{1 + \sin x} = \int \frac{\frac{2\, dt}{1 + t^2}}{1 + \frac{2t}{1 + t^2}}
\]

Simplify the denominator:
\[
1 + \frac{2t}{1 + t^2} = \frac{(1 + t^2) + 2t}{1 + t^2} = \frac{1 + 2t + t^2}{1 + t^2}
\]

Thus, the integral simplifies to:

\[
I = \int \frac{\frac{2\, dt}{1 + t^2}}{\frac{1 + 2t + t^2}{1 + t^2}} = \int \frac{2\, dt}{1 + 2t + t^2}
\]

### Step 3: Complete the square
We can simplify the denominator by completing the square:

\[
1 + 2t + t^2 = (t + 1)^2
\]

So, the integral becomes:

\[
I = \int \frac{2\, dt}{(t + 1)^2}
\]

### Step 4: Solve the integral
The integral of \( \frac{1}{(t + 1)^2} \) is straightforward:

\[
I = \int 2 \cdot (t + 1)^{-2}\, dt = -\frac{2}{t + 1} + C
\]

### Step 5: Back-substitution
Now, we substitute back \( t = \tan\left(\frac{x}{2}\right) \):

\[
I = -\frac{2}{\tan\left(\frac{x}{2}\right) + 1}
\]

### Step 6: Apply limits
Now, we apply the limits \( x = \frac{\pi}{4} \) to \( x = \frac{3\pi}{4} \).

- For \( x = \frac{\pi}{4} \), we have:
  \[
  \tan\left(\frac{\pi}{4} \times \frac{1}{2}\right) = \tan\left(\frac{\pi}{8}\right)
  \]

- For \( x = \frac{3\pi

SOLVE INTEGRAL dx/1+sinx FROM pi/4 TO 3pi/4 BY WRITING SINX IN TERMS OF TAN BUT YOU SHOULD NOT CHANGE LIMITS THAT IT PUT VALUES IN THE FINAL RESULT AFTER RESUBSTITUTING

Learn how to solve the definite integral of 1/(1 + sin(x)) from pi/4 to 3pi/4 by applying Weierstrass substitution and expressing sin(x) in terms of tan(x/2). This solution demonstrates the process of integration using substitution without changing the original limits, making it suitable for undergraduate calculus students.

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