Questions: Simplify the integrand completely. L = ∫ from 0 to 1 sqrt(1 + ( (1-2x) / (2 sqrt(x-x^2)) + 1/(2 sqrt(x) sqrt(1-x)) )^2 ) dx = ∫ from 0 to 1 square dx

Transcript text: Simplify the integrand completely. \[ \begin{aligned} L & =\int_{0}^{1} \sqrt{1+\left(\frac{1-2 x}{2 \sqrt{x-x^{2}}}+\frac{1}{2 \sqrt{x} \sqrt{1-x}}\right)^{2}} d x \\ & =\int_{0}^{1} \square d x \end{aligned} \]

Solution

Solution Steps

To simplify the integrand, we need to combine and simplify the terms inside the square root. This involves algebraic manipulation and simplification of the expression inside the square root.

Step 1: Define the Expression Inside the Square Root

We start with the expression inside the square root: \[ \left(\frac{1 - 2x}{2\sqrt{x - x^2}} + \frac{1}{2\sqrt{x}\sqrt{1 - x}}\right) \]

Step 2: Simplify the Expression

Simplify the expression inside the square root: \[ \frac{1 - 2x}{2\sqrt{x - x^2}} + \frac{1}{2\sqrt{x}\sqrt{1 - x}} = -\frac{x}{\sqrt{x - x^2}} + \frac{1}{2\sqrt{x - x^2}} + \frac{1}{2\sqrt{x}\sqrt{1 - x}} \]

Step 3: Define the Integrand

The integrand is: \[ \sqrt{1 + \left(-\frac{x}{\sqrt{x - x^2}} + \frac{1}{2\sqrt{x - x^2}} + \frac{1}{2\sqrt{x}\sqrt{1 - x}}\right)^2} \]

Step 4: Simplify the Integrand

Simplify the integrand: \[ \sqrt{1 + \left(-\frac{x}{\sqrt{x - x^2}} + \frac{1}{2\sqrt{x - x^2}} + \frac{1}{2\sqrt{x}\sqrt{1 - x}}\right)^2} = \frac{\sqrt{4x^2(1 - x)^2 + \left(\sqrt{x}(1 - 2x)\sqrt{1 - x} + \sqrt{x(1 - x)}\right)^2}}{2x(1 - x)} \]