The value of the integral `int_(0)^(a) (1)/(x+sqrt(a^(2)-x^(2)))dx`, is

Options

A. `pi`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. none of these

Explanation

To solve the integral \( I = \int_{0}^{a} \frac{1}{x + \sqrt{a^2 - x^2}} \, dx \), we will use a trigonometric substitution. Here are the steps:

### Step 1: Substitution
Let \( x = a \sin \theta \). Then, we have:
\[ dx = a \cos \theta \, d\theta \]
Also, the limits change as follows:
- When \( x = 0 \), \( \theta = 0 \)
- When \( x = a \), \( \theta = \frac{\pi}{2} \)

### Step 2: Rewrite the Integral
Substituting \( x \) and \( dx \) into the integral, we get:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{a \cos \theta}{a \sin \theta + \sqrt{a^2 - (a \sin \theta)^2}} \, d\theta \]
Now, simplifying the square root:
\[ \sqrt{a^2 - a^2 \sin^2 \theta} = \sqrt{a^2(1 - \sin^2 \theta)} = \sqrt{a^2 \cos^2 \theta} = a \cos \theta \]
Thus, the integral becomes:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{a \cos \theta}{a \sin \theta + a \cos \theta} \, d\theta \]
Factoring out \( a \):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\cos \theta}{\sin \theta + \cos \theta} \, d\theta \]

### Step 3: Use Symmetry
Now, we can use the identity:
\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]
Let’s consider \( I \) again, but with the substitution \( x = a - x \):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin \theta}{\sin \theta + \cos \theta} \, d\theta \]

### Step 4: Add the Two Integrals
Now we have two expressions for \( I \):
1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\cos \theta}{\sin \theta + \cos \theta} \, d\theta \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin \theta}{\sin \theta + \cos \theta} \, d\theta \)

Adding these two equations:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left( \frac{\cos \theta + \sin \theta}{\sin \theta + \cos \theta} \right) d\theta = \int_{0}^{\frac{\pi}{2}} d\theta \]
This simplifies to:
\[ 2I = \frac{\pi}{2} \]

### Step 5: Solve for \( I \)
Dividing both sides by 2:
\[ I = \frac{\pi}{4} \]

Thus, the value of the integral is:
\[ \boxed{\frac{\pi}{4}} \]