if `int 1/(1+sinx)dx = tan(x/2+a) + C` then find the value of a

Options

A. `a=-pi/4, C in R`
B. `a=pi/4, C in R`
C. `a=(5pi)/(4), C in R`
D. `a=pi/3, C in R`

Answer

To solve the problem, we need to evaluate the integral

\[ \int \frac{1}{1 + \sin x} \, dx \]

and show that it can be expressed in the form \( \tan\left(\frac{x}{2} + a\right) + C \), and then find the value of \( a \).

### Step 1: Rationalizing the Integral

We start with the integral:

\[ \int \frac{1}{1 + \sin x} \, dx \]

To simplify this, we can multiply the numerator and denominator by \( 1 - \sin x \):

\[ \int \frac{1 - \sin x}{(1 + \sin x)(1 - \sin x)} \, dx = \int \frac{1 - \sin x}{1 - \sin^2 x} \, dx \]

### Step 2: Using the Pythagorean Identity

Using the identity \( 1 - \sin^2 x = \cos^2 x \), we rewrite the integral as:

\[ \int \frac{1 - \sin x}{\cos^2 x} \, dx \]

This can be split into two separate integrals:

\[ \int \frac{1}{\cos^2 x} \, dx - \int \frac{\sin x}{\cos^2 x} \, dx \]

### Step 3: Evaluating the Integrals

The first integral \( \int \frac{1}{\cos^2 x} \, dx \) is:

\[ \int \sec^2 x \, dx = \tan x + C_1 \]

The second integral \( \int \frac{\sin x}{\cos^2 x} \, dx \) can be solved using substitution. Let \( u = \cos x \), then \( du = -\sin x \, dx \), which gives:

\[ -\int \frac{1}{u^2} \, du = \frac{1}{\cos x} + C_2 = \sec x + C_2 \]

### Step 4: Combining the Results

Combining the results from the two integrals, we have:

\[ \int \frac{1}{1 + \sin x} \, dx = \tan x - \sec x + C \]

### Step 5: Expressing in Terms of \( \tan\left(\frac{x}{2} + a\right) \)

Now, we need to express \( \tan x - \sec x \) in the form \( \tan\left(\frac{x}{2} + a\right) \).

Using the half-angle formulas:

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

Substituting these into our expression and simplifying will yield:

\[ \tan\left(\frac{x}{2} - \frac{\pi}{4}\right) \]

This implies that:

\[ a = -\frac{\pi}{4} \]

### Final Answer

Thus, the value of \( a \) is:

\[ \boxed{-\frac{\pi}{4}} \]