Question:
$\int\limits^{\pi}_{{0}}[cot\,x]dx, [??$ denotes the greatest integer function, is equal to
$\int\limits^{\pi}_{{0}}[cot\,x]dx, [??$ denotes the greatest integer function, is equal to
Updated On: Jul 28, 2022
- $\frac{\pi}{2}$
- $1$
- $-1$
- $-\frac{\pi}{2}$
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The Correct Option is D
Solution and Explanation
Let $I=\int\limits^{\pi}_{{0}}[cot\,x]dx \,...(1)$
$=\int\limits^{\pi}_{{0}}[cot(\pi-x)]dx\int\limits^{\pi}_{{0}}--cot\,x]dx \,...(1)$
Adding (1) and (2)
$2I=\int\limits^{\pi}_{{0}}[cot\,x]dx+\int\limits^{\pi}_{{0}}[cot\,x]dx=\int\limits^{\pi}_{{0}}(-1)dx\,\,\,$$\left[\because\left[x\right]+\left[-x\right]=-1 \,if\,x\notin Z=0\,if\,x\in Z\right]$
$=\left[-x\right]^{\pi}_{0}=-\pi$
$\therefore I=-\frac{\pi}{2}$
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.