The value of the definite integral `int_(-1)^(1)(1+x)^(1//2)(1-x)^(3//2)dx` equals

Options

A. `pi`
B. `(3pi)/4`
C. `(pi)/4`
D. `(pi)/2`

Explanation

To solve the definite integral

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

we can use the property of definite integrals and symmetry.

### Step 1: Substitute \( x \) with \( -x \)

First, we will consider the substitution \( x = -x \):

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

### Step 2: Write the two integrals

Now we have two expressions for \( I \):

1. \( I = \int_{-1}^{1} (1+x)^{\frac{1}{2}} (1-x)^{\frac{3}{2}} \, dx \) (original integral)
2. \( I = \int_{-1}^{1} (1-x)^{\frac{1}{2}} (1+x)^{\frac{3}{2}} \, dx \) (after substitution)

### Step 3: Add the two integrals

Now, let's add these two integrals:

\[ 2I = \int_{-1}^{1} \left[ (1+x)^{\frac{1}{2}} (1-x)^{\frac{3}{2}} + (1-x)^{\frac{1}{2}} (1+x)^{\frac{3}{2}} \right] \, dx. \]

### Step 4: Factor out common terms

We can factor out the common terms:

\[ 2I = \int_{-1}^{1} \left[ (1+x)^{\frac{1}{2}} (1-x)^{\frac{1}{2}} \left( (1-x) + (1+x) \right) \right] \, dx. \]

### Step 5: Simplify the expression inside the integral

Now simplify the expression inside the integral:

\[ (1-x) + (1+x) = 2. \]

Thus, we have:

\[ 2I = \int_{-1}^{1} 2 (1+x)^{\frac{1}{2}} (1-x)^{\frac{1}{2}} \, dx. \]

### Step 6: Simplify further

This simplifies to:

\[ I = \int_{-1}^{1} (1+x)^{\frac{1}{2}} (1-x)^{\frac{1}{2}} \, dx. \]

### Step 7: Recognize the integral as a standard form

The expression \( (1+x)^{\frac{1}{2}} (1-x)^{\frac{1}{2}} \) can be rewritten as:

\[ \sqrt{(1+x)(1-x)} = \sqrt{1-x^2}. \]

Thus, we have:

\[ I = \int_{-1}^{1} \sqrt{1-x^2} \, dx. \]

### Step 8: Use the known result of the integral

The integral \( \int_{-1}^{1} \sqrt{1-x^2} \, dx \) represents the area of a semicircle with radius 1, which is:

\[ \frac{\pi}{2}. \]

### Final Result

Thus, the value of the definite integral is:

\[ I = \frac{\pi}{2}. \]