51-54 Use the properties of integrals to verify the inequality. ∫π/ 4^π/ 2 (sinx)/(x) d x ⩽(√(2)

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51-54 Use the properties of integrals to verify the...

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Key Concepts

Monotonicity

Monotonicity describes the behavior of a function as either increasing or decreasing over a specific interval. When a function is monotone, it is often easier to determine its maximum or minimum value on the interval, which can then be used to establish bounds on the integral. This concept is particularly useful when the integrand exhibits a consistent increasing or decreasing trend.

Properties of Definite Integrals

These include the linearity and order properties of integrals. The order property states that if one function is less than or equal to another for all points in an interval, then the integral of the first over that interval is similarly bounded by the integral of the second. This concept is essential when comparing or bounding integrals.

Bounding Techniques in Integration

Bounding techniques involve replacing a variable integrand with a constant that is known to be an upper or lower bound on the interval of integration. In practice, if a function is bounded above by a constant over an interval, then the integral is bounded by the product of that constant and the length of the interval. This is a key method in verifying inequalities involving integrals.

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