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

Answer: (A)

4

Answer 2

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