Let `I= int_(0)^(20 20) ( root(3) (x^2) + root(3) ( (1010-x)^(2) ) )/( root(3)(x^(2) ) + 2 root(3) ((1010-x)^(2)) + root(3) ( (2020 - x)^(2) ) ` dx then `(I)/( 100)` is equal to

Answer

To solve the integral

\[ I = \int_{0}^{2020} \frac{\sqrt[3]{x^2} + \sqrt[3]{(1010 - x)^2}}{\sqrt[3]{x^2} + 2\sqrt[3]{(1010 - x)^2} + \sqrt[3]{(2020 - x)^2}} \, dx, \]

we will use the property of definite integrals, which states that

\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx. \]

### Step 1: Apply the property of definite integrals

Let \( a = 0 \) and \( b = 2020 \). We can rewrite the integral as:

\[ I = \int_{0}^{2020} \frac{\sqrt[3]{(2020 - x)^2} + \sqrt[3]{(1010 - (2020 - x))^2}}{\sqrt[3]{(2020 - x)^2} + 2\sqrt[3]{(1010 - (2020 - x))^2} + \sqrt[3]{x^2}} \, dx. \]

### Step 2: Simplify the expression

Now, simplifying the terms inside the integral:

- The first term in the numerator becomes \( \sqrt[3]{(2020 - x)^2} \).
- The second term becomes \( \sqrt[3]{(1010 - (2020 - x))^2} = \sqrt[3]{(x - 1010)^2} \).

Thus, the integral can be rewritten as:

\[ I = \int_{0}^{2020} \frac{\sqrt[3]{(2020 - x)^2} + \sqrt[3]{(x - 1010)^2}}{\sqrt[3]{(2020 - x)^2} + 2\sqrt[3]{(x - 1010)^2} + \sqrt[3]{x^2}} \, dx. \]

### Step 3: Add the two integrals

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

1. \( I = \int_{0}^{2020} \frac{\sqrt[3]{x^2} + \sqrt[3]{(1010 - x)^2}}{\sqrt[3]{x^2} + 2\sqrt[3]{(1010 - x)^2} + \sqrt[3]{(2020 - x)^2}} \, dx \)
2. \( I = \int_{0}^{2020} \frac{\sqrt[3]{(2020 - x)^2} + \sqrt[3]{(x - 1010)^2}}{\sqrt[3]{(2020 - x)^2} + 2\sqrt[3]{(x - 1010)^2} + \sqrt[3]{x^2}} \, dx \)

Adding these two integrals gives:

\[ 2I = \int_{0}^{2020} \frac{\sqrt[3]{x^2} + \sqrt[3]{(1010 - x)^2} + \sqrt[3]{(2020 - x)^2} + \sqrt[3]{(x - 1010)^2}}{\sqrt[3]{x^2} + 2\sqrt[3]{(1010 - x)^2} + \sqrt[3]{(2020 - x)^2}} \, dx. \]

### Step 4: Evaluate the integral

Notice that the numerator and denominator are symmetric and will simplify to:

\[ 2I = \int_{0}^{2020} 1 \, dx = 2020. \]

Thus, we have:

\[ I = \frac{2020}{2} = 1010. \]

### Step 5: Find \( \frac{I}{100} \)

Finally, we need to find:

\[ \frac{I}{100} = \frac{1010}{100} = 10.1. \]

### Final Answer

Thus, the value of \( \frac{I}{100} \) is

\[ \boxed{10.1}. \]