The value of `int(dx)/(x+sqrt(a^(2)-x^(2)))`, is equal to

Options

A. `1/2 sin^(-1)(x/a)+1/2 log |x+sqrt(a^(2)-x^(2))|+C_(1)`
B. `1/2 sin^(-1)(x/a)-1/2log |x+sqrt(a^(2)-x^(2))|+C_(1)`
C. `1/2 sin^(-1)(x/a)-log|x+sqrt(a^(2)-x^(2))|+C_(1)`
D. `1/2 cos^(-1)(x/a)+1/2log|x+sqrt(a^(2)-x^(2))|+C_(1)`

Answer

To solve the integral \( I = \int \frac{dx}{x + \sqrt{a^2 - x^2}} \), we will use a trigonometric substitution. Here are the steps:

### Step 1: Substitution
Let \( x = a \sin \theta \). Then, the differential \( dx \) becomes:
\[ dx = a \cos \theta \, d\theta \]

### Step 2: Substitute in the Integral
Substituting \( x \) and \( dx \) into the integral, we get:
\[ I = \int \frac{a \cos \theta \, d\theta}{a \sin \theta + \sqrt{a^2 - (a \sin \theta)^2}} \]

### Step 3: Simplify the Square Root
The expression under the square root simplifies as follows:
\[ \sqrt{a^2 - (a \sin \theta)^2} = \sqrt{a^2(1 - \sin^2 \theta)} = \sqrt{a^2 \cos^2 \theta} = a \cos \theta \]
Thus, the integral becomes:
\[ I = \int \frac{a \cos \theta \, d\theta}{a \sin \theta + a \cos \theta} \]
This can be simplified further:
\[ I = \int \frac{\cos \theta \, d\theta}{\sin \theta + \cos \theta} \]

### Step 4: Split the Integral
We can rewrite the integrand:
\[ I = \int \frac{\cos \theta}{\sin \theta + \cos \theta} \, d\theta \]
To simplify this, we can add and subtract \( \sin \theta \):
\[ I = \int \frac{\cos \theta + \sin \theta - \sin \theta}{\sin \theta + \cos \theta} \, d\theta = \int \left(1 - \frac{\sin \theta}{\sin \theta + \cos \theta}\right) d\theta \]

### Step 5: Integrate
Now, we can integrate term by term:
\[ I = \int d\theta - \int \frac{\sin \theta}{\sin \theta + \cos \theta} \, d\theta \]
The first integral is simply \( \theta \). For the second integral, we can use the substitution \( u = \sin \theta + \cos \theta \), which gives \( du = (\cos \theta - \sin \theta) d\theta \).

### Step 6: Back Substitution
After integrating and simplifying, we substitute back \( \theta = \sin^{-1} \left( \frac{x}{a} \right) \) and express everything in terms of \( x \):
\[ I = \frac{1}{2} \sin^{-1} \left( \frac{x}{a} \right) + \frac{1}{2} \log \left( x + \sqrt{a^2 - x^2} \right) + C \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{1}{2} \sin^{-1} \left( \frac{x}{a} \right) + \frac{1}{2} \log \left( x + \sqrt{a^2 - x^2} \right) + C \]