Đề bài: Cho hàm số y=f(x) thoả mãn \(\int\limits_0^{\frac{\pi }{2}} {{\mathop{\rm s}\nolimits} {\rm{inx}}\,f(x)dx = f(0) = 1.} \) Tính \(\int\limits_0^{\frac{\pi }{2}} {{\rm{cosx}}\,f'(x)dx.} \)

Đặt \(u = c{\rm{osx;}}\,\,{\rm{v = f(x)}}{\rm{.}}\)

\(I = \int\limits_0^{\frac{\pi }{2}} {{\rm{cos}}} x.f'(x)dx = \left. {c{\rm{os}}x.f(x)} \right|_0^{\frac{\pi }{2}} – \int\limits_0^{\frac{\pi }{2}} {( – {\mathop{\rm s}\nolimits} {\rm{inx)}}{\rm{.f(x)dx}}} \)

\( =  – f(0) + \int\limits_0^{\frac{\pi }{2}} {{\mathop{\rm s}\nolimits} {\rm{inx}}.f(x)dx =  – 1 + 1 = 0} \)