If {x} denotes the fractional part of x, then `I = int _(0) ^(100) (sqrtx)dx,` then the value of `(9I)/(155)` is:

Answer

To solve the integral \( I = \int_{0}^{100} \sqrt{x} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We can express \( \sqrt{x} \) as \( x^{1/2} \). Therefore, we rewrite the integral as:
\[ I = \int_{0}^{100} x^{1/2} \, dx \]

**Hint:** Remember that the integral of \( x^n \) is given by \( \frac{x^{n+1}}{n+1} \).

### Step 2: Apply the Power Rule for Integration
Using the power rule for integration, we have:
\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = \frac{1}{2} \):
\[ I = \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{100} \]

**Hint:** Make sure to apply the limits after finding the antiderivative.

### Step 3: Simplify the Antiderivative
The expression simplifies to:
\[ I = \left[ \frac{2}{3} x^{3/2} \right]_{0}^{100} \]

**Hint:** Substitute the upper and lower limits carefully.

### Step 4: Evaluate the Limits
Now we evaluate the integral from 0 to 100:
\[ I = \frac{2}{3} \left( 100^{3/2} - 0^{3/2} \right) \]
Calculating \( 100^{3/2} \):
\[ 100^{3/2} = (100^{1/2})^3 = 10^3 = 1000 \]
Thus,
\[ I = \frac{2}{3} \cdot 1000 = \frac{2000}{3} \]

**Hint:** Ensure that you correctly compute \( 100^{3/2} \).

### Step 5: Calculate \( \frac{9I}{155} \)
Now we need to find \( \frac{9I}{155} \):
\[ \frac{9I}{155} = \frac{9 \cdot \frac{2000}{3}}{155} \]
This simplifies to:
\[ = \frac{6000}{3 \cdot 155} = \frac{6000}{465} \]

**Hint:** Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor.

### Step 6: Final Simplification
Now, we simplify \( \frac{6000}{465} \):
\[ = \frac{6000 \div 15}{465 \div 15} = \frac{400}{31} \]

Thus, the final answer is:
\[ \frac{9I}{155} = \frac{1200}{31} \]

**Final Answer:** \( \frac{1200}{31} \)