Evaluate the following integral:

dx

Let f(x) =


Substituting x = – x in f(x)


f( – x) =


f(x) = f( – x)


it is an even function


………(1)


Now,


f(x) = |x cosπx| = x cosπx; for x [0,12]


= – x cosπx; for x [1/2,1]


Using interval addition property of integration, we know that



Equation 1 can be written as


2[]


Putting the limits in above equation


= 2{[(x/π)sinx + (1/π2)cosπx]01/2 – [(x/πsinx + (1/π2)cosπx]11/2}


= 2{[(1/2π) – (1/π2)] – [( – 1/π2) – (1/2π)]}


= 2/π


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