\[\frac{d}{d x}\left\{F^{\prime}(x) \sin x-F(x) \cos x\right\}=F^{\prime \prime}(x) \sin x+F(x) \sin x=f(x) \sin x\] and \[ \int_{0}^{\pi} f(x) \sin x d x=\left[F^{\prime}(x) \sin x-F(x) \cos x\right]_{0}^{\pi}=F(\pi)+F(0)\;\;\;\;\;\;\;\text { (1) }\] We have \(F(\pi)+F(0)\) is an integer, since \(f^{(i)}(\pi)\) and \(f^{(i)}(0)\) are integers. However for \(0<x<\pi\), \[0<f(x) \sin x<\frac{\pi^{n} a^{n}}{n !}\] The integral in \(\text { (1) }\) is positive, but arbitrarily small for sufficiently large \(n\) . Therefore \(\text { (1) }\) is false, consequently \[\large \pi \text { is an irrational number }\]