solve this---**Textual Information:**
* **Header/Title:** MIT INTEGRATION BEE 2025 (QUALIFIERS)
* **Question Stem (Mathematical Formula):**
∫₀¹ √(x(1 - x)) dx
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* **Type:** Logo/Emblem
* **Main Elements:**
* Located in the top-left corner of the image.
* Features a stylized white integral symbol (`f`).
* A yellow bee with black stripes, black antennae, and small transparent wings is positioned on the loop of the integral symbol.
* Below the integral symbol, the text "MIT" is stacked above "2025" in white, sans-serif font.
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Welcome to the MIT Integration Bee 2025! We need to evaluate the integral from 0 to 1 of the square root of x times 1 minus x. This function represents a semicircle-like curve, and we're finding the area under this curve. The integrand has its maximum at x equals one half, where it equals one half. This suggests we might use a trigonometric substitution to solve it.
To solve this integral, we use trigonometric substitution. First, we recognize that x times 1 minus x can be rewritten as x minus x squared. Then we complete the square to get one fourth minus x minus one half squared. This is the equation of a semicircle with radius one half. We substitute x minus one half equals one half sine theta, which transforms our integral into a trigonometric form that's easier to evaluate.
Now we apply our substitution. The original integral becomes an integral from negative pi over 2 to pi over 2. Under the square root, we get one fourth minus one fourth sine squared theta, which simplifies to one half cosine theta. Combined with the dx term, we get one fourth times the integral of cosine squared theta. This is a standard trigonometric integral that we can evaluate using the cosine squared identity.