`int_(0)^(4)(3sqrt(x+5))/(3sqrt(x+5)+3sqrt(9-x))dx=`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the integral

\[ I = \int_{0}^{4} \frac{3\sqrt{x+5}}{3\sqrt{x+5} + 3\sqrt{9-x}} \, dx, \]

we can simplify the problem using a symmetry property of definite integrals.

### Step 1: Use the substitution \( x = 4 - t \)

Let \( x = 4 - t \). Then, \( dx = -dt \). When \( x = 0 \), \( t = 4 \) and when \( x = 4 \), \( t = 0 \). Thus, the integral becomes:

\[ I = \int_{4}^{0} \frac{3\sqrt{(4-t)+5}}{3\sqrt{(4-t)+5} + 3\sqrt{9-(4-t)}} (-dt) = \int_{0}^{4} \frac{3\sqrt{9-t}}{3\sqrt{9-t} + 3\sqrt{t+5}} \, dt. \]

### Step 2: Rewrite the integral

Now we have:

\[ I = \int_{0}^{4} \frac{3\sqrt{9-t}}{3\sqrt{9-t} + 3\sqrt{t+5}} \, dt. \]

### Step 3: Add the two integrals

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

1. \( I = \int_{0}^{4} \frac{3\sqrt{x+5}}{3\sqrt{x+5} + 3\sqrt{9-x}} \, dx \)
2. \( I = \int_{0}^{4} \frac{3\sqrt{9-x}}{3\sqrt{9-x} + 3\sqrt{x+5}} \, dx \)

Adding these two integrals:

\[ 2I = \int_{0}^{4} \left( \frac{3\sqrt{x+5}}{3\sqrt{x+5} + 3\sqrt{9-x}} + \frac{3\sqrt{9-x}}{3\sqrt{9-x} + 3\sqrt{x+5}} \right) dx. \]

### Step 4: Simplify the expression

The sum of the fractions simplifies as follows:

\[ \frac{3\sqrt{x+5}}{3\sqrt{x+5} + 3\sqrt{9-x}} + \frac{3\sqrt{9-x}}{3\sqrt{9-x} + 3\sqrt{x+5}} = 1. \]

Thus, we have:

\[ 2I = \int_{0}^{4} 1 \, dx. \]

### Step 5: Evaluate the integral

Now, we can evaluate the integral:

\[ \int_{0}^{4} 1 \, dx = [x]_{0}^{4} = 4 - 0 = 4. \]

### Step 6: Solve for \( I \)

Therefore, we have:

\[ 2I = 4 \implies I = \frac{4}{2} = 2. \]

### Final Answer

Thus, the value of the integral is:

\[ \boxed{2}. \]