Not so standard

Calculus Level 5

Evaluate: \[ \int \limits_0^\infty ( 2 + f(x) )(1- f(x) ) \frac{ \text{d}x}{x^2} \] where, \(\displaystyle f(x) = \frac{ \sin x }{x} \).

The value of the integral can be expressed as \(\displaystyle \frac{a \pi ^b }{c} \).

Given \(a\) and \(c\) are coprime, submit the value of \( a+ b+ c \).

Details and Assumptions:
Following integrals may be helpful:

  • \(\displaystyle \int \limits_0^\infty \big( f(x) \big)^2 \text{d}x = \frac{ \pi }{2} \)

  • \(\displaystyle \int \limits_0^\infty \big( f(x) \big)^3 \text{d}x = \frac{3 \pi }{8} \)

  • \(\displaystyle \int \limits_0^\infty \big( f(x) \big)^4 \text{d}x = \frac{ \pi }{3} \)

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