Question

For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [–

Answer 1

The question is incomplete here is the complete question:

For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined to the interval [−10,10] by

f(x)={x−[x]1+[x]−xif [x] is oddif [x] is even

Then the value of π210∫−1010f(x)cosπx dx is

Answer:

f(x)={{x}1−{x}2n−1≤x<2n2n≤x<2n+1

f(x) is a periodic function with period = 2

f(x).cosπx  is also periodic with period = 2

π210∫−1010f(x)cos(πx)dx=π2∫02f(x)cos(πx)dx=π2∫01((1−{x})+{−x})cos(πx)dx

2π2∫01(−xcosπx)dx = − 2π2[xsinπxπ+cosπxπ2]10=−2π2(−2π2)=4  

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