Let \([t]\) denote the greatest integer less than or equal to \(t\). Then, the value of the integral \(∫_0^1 [ -8x^2 + 6x - 1] dx\) is equal to
\(-1\)
\(-\frac 54\)
\(\frac {\sqrt {17}-13}{8}\)
\(\frac {\sqrt {17}-16}{8}\)
The Correct Option is C
Solution and Explanation
![Let [t] denote the greatest integer less than or equal to t](https://images.collegedunia.com/public/qa/images/content/2024_01_18/Screenshot_1b4c1fc71705587192253.png)
\(∫_0^1 [ -8x^2 + 6x - 1] dx\)
\(= ∫_0^\frac 14 (-1)dx + ∫_{\frac 14}^{\frac 34} 0dx + ∫_{\frac 34}^{\frac 12} (-1)dx + ∫_{\frac {3+\sqrt {17}}{8}}^{\frac 34} (-1)dx + ∫_{\frac {3+\sqrt {17}}{8}}^{1}(-3)dx\)
\(= -\frac 14 -\frac 14 - 2 ( \frac {3+\sqrt {17}}{8} - \frac 34) - 3 (1 - \frac {3+\sqrt {17}}{8})\)
\(=\frac {\sqrt {17}-13}{8}\)
So, the correct option is (C): \(\frac {\sqrt {17}-13}{8}\)
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Concepts Used:
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus is the theorem which states that differentiation and integration are opposite processes (or operations) of one another.
Calculus's fundamental theorem connects the notions of differentiating and integrating functions. The first portion of the theorem - the first fundamental theorem of calculus – asserts that by integrating f with a variable bound of integration, one of the antiderivatives (also known as an indefinite integral) of a function f, say F, can be derived. This implies the occurrence of antiderivatives for continuous functions.