Evaluate the following integral: ∫ |x cos π x| dx, x ∈ [-1, 1]

Question

Evaluate the following integral:

\(\int\limits_{-1}^{1} \) |x cos π x| dx

Answer 1

Let f(x) =   |x cos π x|

Substituting x = -x in f(x)

Now,

f(x) = |x cosπx| = x cosπx; for x ∈ [0,1/2]

= – x cosπx; for x [1/2,1]

Using interval addition property of integration, we know that

Putting the limits in above equation

= 2{[(x/π)sin x + (1/π²)cosπx]0 ^1/2 – [(x/π sin x + (1/π²)cosπx]¹ _1/2}

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

= 2/π