Mathematics

Mathematics

Problem

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 7. ∫01x(1−x)ndx\int_{0}^{1} x(1-x)^{n} d x
Intuitive explanation

This integral asks you to find the area under a product of a linear function and a power function. The key insight is to use integration by parts strategically: treat xx as the part to differentiate (so it vanishes) and (1−x)n(1-x)^n as the part to integrate. Alternatively, you can recognize this as a Beta function or use a substitution like u=1−xu = 1-x to simplify the form. Either way, you're converting a product into something more manageable.

Intuitive explanation

This integral asks you to find the area under a product of a linear function and a power function. The key insight is to use integration by parts strategically: treat xx as the part to differentiate (so it vanishes) and (1−x)n(1-x)^n as the part to integrate. Alternatively, you can recognize this as a Beta function or use a substitution like u=1−xu = 1-x to simplify the form. Either way, you're converting a product into something more manageable.

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