The value of the integral `int_(0)^(1) (1)/((1+x^(2))^(3//2))dx` is

Options

A. `1//2`
B. `1//sqrt(2)`
C. 1
D. `sqrt(2)`

Answer

To solve the integral

\[ I = \int_{0}^{1} \frac{1}{(1 + x^2)^{3/2}} \, dx, \]

we will use the substitution \( x = \tan(\theta) \).

### Step 1: Substitution
Let \( x = \tan(\theta) \). Then, the differential \( dx \) becomes:

\[ dx = \sec^2(\theta) \, d\theta. \]

### Step 2: Change of Limits
When \( x = 0 \):
\[ \theta = \tan^{-1}(0) = 0. \]
When \( x = 1 \):
\[ \theta = \tan^{-1}(1) = \frac{\pi}{4}. \]

### Step 3: Rewrite the Integral
Now, we can rewrite the integral in terms of \( \theta \):

\[ I = \int_{0}^{\frac{\pi}{4}} \frac{\sec^2(\theta)}{(1 + \tan^2(\theta))^{3/2}} \, d\theta. \]

Using the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \), we have:

\[ I = \int_{0}^{\frac{\pi}{4}} \frac{\sec^2(\theta)}{(\sec^2(\theta))^{3/2}} \, d\theta. \]

### Step 4: Simplify the Integral
This simplifies to:

\[ I = \int_{0}^{\frac{\pi}{4}} \frac{\sec^2(\theta)}{\sec^3(\theta)} \, d\theta = \int_{0}^{\frac{\pi}{4}} \cos(\theta) \, d\theta. \]

### Step 5: Integrate
Now we can integrate \( \cos(\theta) \):

\[ I = \int_{0}^{\frac{\pi}{4}} \cos(\theta) \, d\theta = \left[ \sin(\theta) \right]_{0}^{\frac{\pi}{4}}. \]

### Step 6: Evaluate the Limits
Evaluating the limits gives:

\[ I = \sin\left(\frac{\pi}{4}\right) - \sin(0) = \frac{1}{\sqrt{2}} - 0 = \frac{1}{\sqrt{2}}. \]

Thus, the value of the integral is

\[ \boxed{\frac{1}{\sqrt{2}}}. \]