Let f be a real-valued function of a real variable defined as f(x) = x – [x], where [x] denotes the largest integer less than or equal to x. The value - General SHORT_ANSWER PYQs Based Test - 04 : Definite and Indefinite Integrals | ExamTest.live

mediumSHORT_ANSWERPYQs Based Test - 04 : Definite and Indefinite IntegralsGeneral

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Let f be a real-valued function of a real variable defined as f(x) = x - [x], where [x] denotes the largest integer less than or equal to x. The value of 0.25 \int 1.25 f (x) d x is _______ (up to 2 decimal places).

Correct Answer

Solution & Step-by-step Explanation

In the range: 0.49 - 0.51 f(x) = x - [x] for 0.25 < x < 1, f(x) = x for 1 < x < 1.25, f(x) = x - 1 0.25 \int 1.25 f (x) d x = 0.25 \int 1 x d x + 1 \int 1.25 (x - 1) d x = [\frac{x ^{2}}{2}]_{0.25}^{1} + [\frac{x ^{2}}{2} - x]_{1}^{1.25} = \frac{1}{2} [1 - \frac{1}{16}] + [(\frac{25}{32} - \frac{5}{4}) - (\frac{1}{2} - 1)] = \frac{1}{2} [\frac{15}{16}] + [\frac{25}{32} - \frac{5}{4} + \frac{1}{2}] = \frac{15 + 25 - 40 + 16}{32} = 0.5

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Let f be a real-valued function of a real variable defined as f(x) = x - [x], where [x] denotes the largest integer less than or equal to x. The value of 0.25 \int 1.25 f (x) d x is _______ (up to 2 decimal places).