Evaluate the following integrals:
`int(x)/((1+sinx))dx`

Explanation

To evaluate the integral \( I = \int \frac{x}{1 + \sin x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express the integral in a different form:
\[ I = \int \frac{x}{1 + \sin x} \, dx = \int \frac{x \cos\left(\frac{\pi}{2} - x\right)}{1 + \sin x} \, dx \]

### Step 2: Use the Identity
Using the identity \( 1 + \sin x = 1 + \cos\left(\frac{\pi}{2} - x\right) \), we can rewrite the integral as:
\[ I = \int \frac{x}{1 + \cos\left(\frac{\pi}{2} - x\right)} \, dx \]

### Step 3: Apply the Half Angle Formula
Using the formula \( 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \):
\[ 1 + \sin x = 2 \cos^2\left(\frac{\pi/4 - x/2}{2}\right) \]
So we rewrite the integral:
\[ I = \int \frac{x}{2 \cos^2\left(\frac{\pi}{4} - \frac{x}{2}\right)} \, dx \]

### Step 4: Factor Out the Constant
Factoring out the constant:
\[ I = \frac{1}{2} \int x \sec^2\left(\frac{\pi}{4} - \frac{x}{2}\right) \, dx \]

### Step 5: Integration by Parts
Let \( u = x \) and \( dv = \sec^2\left(\frac{\pi}{4} - \frac{x}{2}\right) \, dx \).
Then, \( du = dx \) and we need to find \( v \):
\[ v = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \]

### Step 6: Apply Integration by Parts Formula
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ I = \frac{1}{2} \left[ x \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) - \int \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \, dx \right] \]

### Step 7: Evaluate the Remaining Integral
To evaluate \( \int \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \, dx \), we can use the substitution \( t = \frac{\pi}{4} - \frac{x}{2} \):
\[ dx = -2 \, dt \]
Thus,
\[ \int \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \, dx = -2 \int \tan(t) \, dt = -2 \log|\sec(t)| + C \]

### Step 8: Substitute Back
Substituting back for \( t \):
\[ -2 \log|\sec\left(\frac{\pi}{4} - \frac{x}{2}\right)| + C \]

### Final Answer
Combining everything, we have:
\[ I = \frac{1}{2} \left[ x \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) + 2 \log|\sec\left(\frac{\pi}{4} - \frac{x}{2}\right)| \right] + C \]

### Final Result
Thus, the evaluated integral is:
\[ I = -x \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) + \log|\sec\left(\frac{\pi}{4} - \frac{x}{2}\right)| + C \]