Not so standard

Calculus Level 5

Evaluate: 0(2+f(x))(1f(x))dxx2 \int \limits_0^\infty ( 2 + f(x) )(1- f(x) ) \frac{ \text{d}x}{x^2} where, f(x)=sinxx\displaystyle f(x) = \frac{ \sin x }{x} .

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

Given aa and cc are coprime, submit the value of a+b+c a+ b+ c .

Details and Assumptions:
Following integrals may be helpful:

  • 0(f(x))2dx=π2\displaystyle \int \limits_0^\infty \big( f(x) \big)^2 \text{d}x = \frac{ \pi }{2}

  • 0(f(x))3dx=3π8\displaystyle \int \limits_0^\infty \big( f(x) \big)^3 \text{d}x = \frac{3 \pi }{8}

  • 0(f(x))4dx=π3\displaystyle \int \limits_0^\infty \big( f(x) \big)^4 \text{d}x = \frac{ \pi }{3}


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