Biết \(\int\limits_0^{\frac{\pi }{2}} {f\left( {\sin x} \right){\rm{cos(}}x + \pi ){\rm{d}}x} = - 2\). Tính tích phân \(I = \int\limits_0^1 {f\left( x \right){\rm{d}}x} \).

+ Ta có: \(\int\limits_0^{\frac{\pi }{2}} {f\left( {\sin x} \right){\rm{cos(}}x + \pi ){\rm{d}}x} \) \( = - \int\limits_0^{\frac{\pi }{2}} {f\left( {\sin x} \right){\rm{cos}}x{\rm{d}}x} = - 2\).

+ Suy ra \(\int\limits_0^{\frac{\pi }{2}} {f\left( {\sin x} \right){\rm{cos}}x{\rm{d}}x} = 2\).

+ Đặt : \(t = \sin x \Rightarrow {\rm{d}}t = {\rm{cos}}x{\rm{d}}x\), khi đó:

\(\int\limits_0^{\frac{\pi }{2}} {f\left( {\sin x} \right){\rm{cos}}x{\rm{d}}x} \)\( = \int\limits_0^1 {f\left( t \right){\rm{d}}t} \)\({\rm{ = }}\int\limits_0^1 {f\left( x \right){\rm{d}}x} \)\( \Rightarrow I = \int\limits_0^1 {f\left( x \right){\rm{d}}x} = 2\).