\int _0^{100}\frac{1}{200}\cdot \left(1-\frac{x}{100}\right)^{-\frac{1}{2}}\cdot \:x^2\:dx

To solve the integral

\[
\int _0^{100}\frac{1}{200}\cdot \left(1-\frac{x}{100}\right)^{-\frac{1}{2}}\cdot \:x^2\:dx,
\]

we will follow these steps:

### Step 1: Simplify the integral expression
First, rewrite the integrand:

\[
\int_0^{100} \frac{1}{200} \cdot \left(1 - \frac{x}{100}\right)^{-\frac{1}{2}} \cdot x^2 \, dx
\]

This can be factored out:

\[
\frac{1}{200} \int_0^{100} x^2 \left(1 - \frac{x}{100}\right)^{-\frac{1}{2}} \, dx
\]

### Step 2: Perform a substitution
Let \( u = 1 - \frac{x}{100} \), then:

\[
du = -\frac{1}{100} dx \quad \text{or} \quad dx = -100 \, du
\]

Also, \( x = 100(1 - u) \).

### Step 3: Change the limits of integration
When \( x = 0 \), \( u = 1 \), and when \( x = 100 \), \( u = 0 \).

### Step 4: Substitute and simplify the integral
Substitute into the integral:

\[
\frac{1}{200} \int_1^0 (100(1 - u))^2 \cdot u^{-\frac{1}{2}} \cdot (-100) \, du
\]

Simplify the expression:

\[
\frac{1}{200} \cdot 100^3 \int_0^1 (1 - u)^2 u^{-\frac{1}{2}} \, du
\]

\[
\frac{1000000}{200} \int_0^1 (1 - u)^2 u^{-\frac{1}{2}} \, du
\]

\[
5000 \int_0^1 (1 - u)^2 u^{-\frac{1}{2}} \, du
\]

### Step 5: Expand the integrand
Expand \((1 - u)^2\):

\[
(1 - u)^2 = 1 - 2u + u^2
\]

So the integral becomes:

\[
5000 \int_0^1 (1 - 2u + u^2) u^{-\frac{1}{2}} \, du
\]

This can be split into three integrals:

\[
5000 \left(\int_0^1 u^{-\frac{1}{2}} \, du - 2\int_0^1 u^{\frac{1}{2}} \, du + \int_0^1 u^{\frac{3}{2}} \, du\right)
\]

### Step 6: Evaluate each integral separately
1. **First Integral**: \(\int_0^1 u^{-\frac{1}{2}} \, du\)

   \[
   \int_0^1 u^{-\frac{1}{2}} \, du = \left[ 2u^{\frac{1}{2}} \right]_0^1 = 2 \cdot (1^{\frac{1}{2}} - 0^{\frac{1}{2}}) = 2
   \]

2. **Second Integral**: \(\int_0^1 u^{\frac{1}{2}} \, du\)

   \[
   \int_0^1 u^{\frac{1}{2}} \, du = \left[ \frac{2}{3} u^{\frac{3}{2}} \right]_0^1 = \frac{2}{3}
   \]

3. **Third Integral**: \(\int_0^1 u^{\frac{3}{2}} \, du\)

   \[
   \int_0^1 u^{\frac{3}{2}} \, du = \left[ \frac{2}{5} u^{\frac{5}{2}} \right]_0^1 = \frac{2}{5}
   \]

### Step 7: Combine the results
Now, substitute the results into the expression:

\[
5000 \left(2 - 2 \cdot \frac{2}{3} + \frac{2}{5}\right)
\]

Simplify inside the parentheses:

\[
5000 \left(\frac{6}{3} - \frac{4}{3} + \frac{2}{5}\right) = 5000 \left(\frac{2}{3} + \frac{2}{5}\right)
\]

Find a common denominator:

\[
\frac{2}{3} + \frac{2}{5} = \frac{10}{15} + \frac{6}{15} = \frac{16}{15}
\]

So, the integral becomes:

\[
5000 \cdot \frac{16}{15} = \frac{80000}{15} = \frac{16000}{3} \approx 5333.33
\]

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

\[
\frac{16000}{3} \approx 5333.33
\]

Would you like more details or have any questions?

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Here are some related questions:
1. What is the impact of changing the exponent on \( x^2 \) in the integrand?
2. How does changing the limits of integration affect the result?
3. What other substitution methods could simplify this integral?
4. How can this integration technique be applied to other forms of polynomials?
5. What is the significance of the integral's result in terms of area under the curve?

**Tip:** When integrating functions with powers of \( x \), consider expanding and integrating term by term to simplify the process.

Learn how to solve the integral ∫[0 to 100] (1 - x/100)^(-1/2) * x^2 dx step-by-step. This problem involves advanced techniques in integral calculus, including the substitution method and integration by parts. Suitable for advanced college-level mathematics.

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