The value if definite integral `int_(3/2)^(9/4)[sqrt(2x-sqrt(5(4x-5)))+sqrt(2x+sqrt(5(4x-5)))]` dx is equal to

Answer

To solve the definite integral

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

we will follow these steps:

### Step 1: Simplify the Expression Inside the Integral
We start by rewriting the expression under the integral. Notice that:

\[ \sqrt{5(4x - 5)} = \sqrt{20x - 25}. \]

Thus, we can rewrite the integral as:

\[ I = \int_{\frac{3}{2}}^{\frac{9}{4}} \left[\sqrt{2x - \sqrt{20x - 25}} + \sqrt{2x + \sqrt{20x - 25}}\right] dx. \]

### Step 2: Substitution
Let us make the substitution:

\[ t^2 = 5(4x - 5). \]

This implies:

\[ 4x - 5 = \frac{t^2}{5} \implies 4x = \frac{t^2}{5} + 5 \implies x = \frac{t^2 + 25}{20}. \]

Now, differentiate both sides to find \(dx\):

\[ dx = \frac{1}{20} \cdot 2t \, dt = \frac{t}{10} \, dt. \]

### Step 3: Change the Limits of Integration
Next, we need to change the limits of integration.

1. For \(x = \frac{3}{2}\):

\[ t^2 = 5(4 \cdot \frac{3}{2} - 5) = 5(6 - 5) = 5 \implies t = \sqrt{5}. \]

2. For \(x = \frac{9}{4}\):

\[ t^2 = 5(4 \cdot \frac{9}{4} - 5) = 5(9 - 5) = 20 \implies t = \sqrt{20} = 2\sqrt{5}. \]

Thus, the new limits are from \(\sqrt{5}\) to \(2\sqrt{5}\).

### Step 4: Substitute Back into the Integral
Now substituting \(x\) and \(dx\) into the integral, we have:

\[ I = \int_{\sqrt{5}}^{2\sqrt{5}} \left[\sqrt{2\left(\frac{t^2 + 25}{20}\right) - t} + \sqrt{2\left(\frac{t^2 + 25}{20}\right) + t}\right] \cdot \frac{t}{10} \, dt. \]

### Step 5: Simplify the Integral
Now we simplify the expression inside the integral:

\[ 2\left(\frac{t^2 + 25}{20}\right) = \frac{t^2 + 25}{10}. \]

Thus, we have:

\[ I = \int_{\sqrt{5}}^{2\sqrt{5}} \left[\sqrt{\frac{t^2 + 25 - 10t}{10}} + \sqrt{\frac{t^2 + 25 + 10t}{10}}\right] \cdot \frac{t}{10} \, dt. \]

### Step 6: Further Simplification
This can be simplified to:

\[ I = \frac{1}{10} \int_{\sqrt{5}}^{2\sqrt{5}} \left[\sqrt{(t - 5)^2} + \sqrt{(t + 5)^2}\right] \cdot t \, dt. \]

### Step 7: Evaluate the Integral
Now, we can evaluate the integral:

1. \(\sqrt{(t - 5)^2} = |t - 5|\) and \(\sqrt{(t + 5)^2} = |t + 5|\).
2. For \(t\) in \([\sqrt{5}, 2\sqrt{5}]\), \(t - 5\) is negative and \(t + 5\) is positive.

Thus, we have:

\[ I = \frac{1}{10} \int_{\sqrt{5}}^{2\sqrt{5}} \left[-(t - 5) + (t + 5)\right] t \, dt = \frac{1}{10} \int_{\sqrt{5}}^{2\sqrt{5}} 10t \, dt. \]

### Step 8: Final Calculation
Now we compute:

\[ I = \int_{\sqrt{5}}^{2\sqrt{5}} t \, dt = \left[\frac{t^2}{2}\right]_{\sqrt{5}}^{2\sqrt{5}} = \frac{(2\sqrt{5})^2}{2} - \frac{(\sqrt{5})^2}{2} = \frac{20}{2} - \frac{5}{2} = \frac{15}{2}. \]

Thus,

\[ I = \frac{1}{10} \cdot 15 = \frac{3}{2}. \]

### Final Answer
The value of the definite integral is:

\[ \boxed{\frac{3}{2}}. \]