Question

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

• A. 4
• B. 2
• C. 1
• D. 0

Answer 1

The Correct Option is A

\(f(x)=\left\{\begin{matrix}  x=[x] & if \,[x]\, is \,odd\\   1+[x]-x\,& if\,[x] \,is\,even \end{matrix}\right.\)
Graph of f(x)

So,
\(\frac{\pi^2}{10}\int_{-10}^{10}f(x)cos\pi x dx\)=\(\frac{\pi^2}{10}\)⋅20\(\int_{0}^{1}\)f(x)cos⁡πx dx
=2\(\pi^2\)\(\int_{0}^{1}\)(1−x)cos⁡πx dx
=2\(\pi^2\){(1−x)\(\frac{sin⁡πx}{x}\)|⁰₁−\(\frac{cosπx}{\pi^2}\)|⁰₁}=4