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

To solve the integral 

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

we can follow these steps:

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

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

can be simplified as:

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

### 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:

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

Simplify the expression:

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

\[
\frac{10000}{200} \int_0^1 (1 - u) u^{-\frac{1}{2}} \, du
\]

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

### Step 5: Break down the integral
Now, expand the integral:

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

### 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**: \(\int_0^1 u^{\frac{1}{2}} \, du\)

   \[
   \int_0^1 u^{\frac{1}{2}} \, du = \left[ \frac{2}{3} u^{\frac{3}{2}} \right]_0^1 = \frac{2}{3} \cdot (1^{\frac{3}{2}} - 0^{\frac{3}{2}}) = \frac{2}{3}
   \]

### Step 7: Combine the results
Now, subtract the second integral from the first:

\[
50 \left(2 - \frac{2}{3}\right) = 50 \left(\frac{6}{3} - \frac{2}{3}\right) = 50 \cdot \frac{4}{3} = \frac{200}{3}
\]

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

\[
\frac{200}{3} \approx 66.67
\]

Would you like more details or have any questions?

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Here are some related questions:
1. What is the geometric interpretation of this integral?
2. How does substitution simplify complex integrals?
3. What are other methods to solve similar integrals without substitution?
4. How can we apply this technique to a definite integral over different limits?
5. What happens to the integral if the exponent in the integrand changes?

**Tip:** Always consider a substitution when the integrand includes a term that complicates direct integration.

Learn how to solve the integral from 0 to 100 of (1 - x/100)^(-1/2) * x dx using substitution method. Follow a detailed step-by-step solution, including the application of the substitution formula for integrals. Suitable for advanced college-level students.

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