`int_(0)^(a)(a+x)/(sqrt(a-x))dx=`

Options

A. `(a sqrt(a))/(3)`
B. `(5a sqrt(a))/(3)`
C. `(4a sqrt(a))/(3)`
D. `(10a sqrt(a))/(3)`

Answer

To solve the integral \( I = \int_{0}^{a} \frac{a+x}{\sqrt{a-x}} \, dx \), we can use the property of definite integrals. Let's go through the solution step by step.

### Step 1: Define the Integral
Let
\[ I = \int_{0}^{a} \frac{a+x}{\sqrt{a-x}} \, dx \]

### Step 2: Use the Property of Definite Integrals
According to the property of definite integrals, we can express:
\[ I = \int_{0}^{a} \frac{a + (a - x)}{\sqrt{a - (a - x)}} \, dx \]
This simplifies to:
\[ I = \int_{0}^{a} \frac{2a - x}{\sqrt{x}} \, dx \]

### Step 3: Rewrite the Integral
Now we can rewrite \( I \) as:
\[ I = \int_{0}^{a} \frac{2a}{\sqrt{x}} \, dx - \int_{0}^{a} \frac{x}{\sqrt{x}} \, dx \]
This simplifies to:
\[ I = 2a \int_{0}^{a} x^{-1/2} \, dx - \int_{0}^{a} x^{1/2} \, dx \]

### Step 4: Evaluate the Integrals
Now we will evaluate each integral separately.

1. **First Integral**:
\[ \int_{0}^{a} x^{-1/2} \, dx = \left[ 2x^{1/2} \right]_{0}^{a} = 2\sqrt{a} - 0 = 2\sqrt{a} \]

2. **Second Integral**:
\[ \int_{0}^{a} x^{1/2} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_{0}^{a} = \frac{2}{3} a^{3/2} - 0 = \frac{2}{3} a^{3/2} \]

### Step 5: Substitute Back
Now substituting back into our expression for \( I \):
\[ I = 2a(2\sqrt{a}) - \frac{2}{3} a^{3/2} \]
This simplifies to:
\[ I = 4a\sqrt{a} - \frac{2}{3} a^{3/2} \]

### Step 6: Combine the Terms
To combine the terms, we can express \( 4a\sqrt{a} \) as \( \frac{12}{3} a^{3/2} \):
\[ I = \frac{12}{3} a^{3/2} - \frac{2}{3} a^{3/2} = \frac{10}{3} a^{3/2} \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{10}{3} a^{3/2} \]