`int(xcostheta+1)/((x^(2)+2xcostheta+1)^((3)/(2)))dx` is

Options

A. `sqrt(xcostheta+1)+C`
B. `(1)/(sqrt(x^(2)2xcostheta+1))+C`
C. `(x)/(x^(2)+2xcostheta+1))+C`
D. `sqrt(x^(2)+2xcostheta+1)+C`

Explanation

To solve the integral

\[ I = \int \frac{x \cos \theta + 1}{(x^2 + 2x \cos \theta + 1)^{3/2}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Denominator
We can rewrite the denominator \(x^2 + 2x \cos \theta + 1\) as \((x + \cos \theta)^2 + \sin^2 \theta\). This helps us recognize it as a perfect square.

### Step 2: Rewrite the Integral
Now, we can rewrite the integral as:

\[ I = \int \frac{x \cos \theta + 1}{((x + \cos \theta)^2 + \sin^2 \theta)^{3/2}} \, dx. \]

### Step 3: Substitution
Let \(t = x + \cos \theta\), then \(dx = dt\) and \(x = t - \cos \theta\). Substitute these into the integral:

\[ I = \int \frac{(t - \cos \theta) \cos \theta + 1}{(t^2 + \sin^2 \theta)^{3/2}} \, dt. \]

### Step 4: Expand the Numerator
Now expand the numerator:

\[ (t - \cos \theta) \cos \theta + 1 = t \cos \theta - \cos^2 \theta + 1. \]

### Step 5: Combine Terms
The numerator can be rewritten as:

\[ t \cos \theta + (1 - \cos^2 \theta) = t \cos \theta + \sin^2 \theta. \]

### Step 6: Separate the Integral
Now we can separate the integral into two parts:

\[ I = \int \frac{t \cos \theta}{(t^2 + \sin^2 \theta)^{3/2}} \, dt + \int \frac{\sin^2 \theta}{(t^2 + \sin^2 \theta)^{3/2}} \, dt. \]

### Step 7: Solve Each Integral
1. For the first integral, use the substitution \(u = t^2 + \sin^2 \theta\).
2. For the second integral, we can use a standard integral formula.

### Step 8: Combine Results
After integrating both parts, we will combine the results and substitute back \(t = x + \cos \theta\).

### Step 9: Final Result
The final result will be expressed in terms of \(x\) and \(\theta\).

### Conclusion
After performing all the steps, we find that the integral evaluates to:

\[ \frac{1}{\sqrt{1 + 2 \cos \theta + \frac{1}{x^2}}} + C. \]

Thus, the correct answer is:

**Option C: \(\frac{1}{\sqrt{1 + 2 \cos \theta + \frac{1}{x^2}}} + C\)**.