For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real-valued function defined on the interval [–10, 10] by \(f(x)=\left\{\begin{matrix} x=[x] & if \,[x]\, is \,odd\\ 1+[x]-x\,& if\,[x] \,is\,even \end{matrix}\right.\), if [x] is even, the value of \(\frac{\pi^2}{10}\int_{-10}^{10}f(x)cos\pi x dx\) is
- 4
- 2
- 1
- 0
The Correct Option is A
Solution and Explanation
\(f(x)=\left\{\begin{matrix} x=[x] & if \,[x]\, is \,odd\\ 1+[x]-x\,& if\,[x] \,is\,even \end{matrix}\right.\)
Graph of f(x)![[x] denote the largest integer less than equal to x](https://images.collegedunia.com/public/qa/images/content/2024_01_04/jeemain202_04102d001704371092614.png)
So,
\(\frac{\pi^2}{10}\int_{-10}^{10}f(x)cos\pi x dx\)=\(\frac{\pi^2}{10}\)⋅20\(\int_{0}^{1}\)f(x)cosπx dx
=2\(\pi^2\)\(\int_{0}^{1}\)(1−x)cosπx dx
=2\(\pi^2\){(1−x)\(\frac{sinπx}{x}\)|01−\(\frac{cosπx}{\pi^2}\)|01}=4
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).