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* **Event Title:** MIT INTEGRATION BEE 2025 (QUALIFIERS)
* **Logo Text:** MIT 2025 (accompanying a stylized 'f' symbol with a bee)
**Question Stem (Mathematical Formula):**
The question asks to evaluate the definite integral:
∫ from 0 to 1 of √(x(1 - x)) dx
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Today we're solving a problem from the MIT Integration Bee 2025 qualifiers. We need to evaluate the definite integral from 0 to 1 of the square root of x times 1 minus x. The integrand involves a square root of a quadratic expression x times 1 minus x, which makes direct integration quite challenging. We'll need to use a clever substitution approach to solve this integral.
To solve this integral, we need to set up a trigonometric substitution. First, let's analyze the expression x times 1 minus x, which equals x minus x squared. We can complete the square to get one quarter minus x minus one half squared. This form suggests a semicircle with radius one half. We choose the substitution x equals one half plus one half sine theta, which gives us dx equals one half cosine theta d theta. The limits transform from x in 0 to 1 to theta in negative pi over 2 to pi over 2.