Cho hàm số . Tính \(I = 2\int_0^{\frac{\pi }{2}} {f\left( {\sin x} \right)\cos x{\rm{d}}x} + 3\int_0^1 {f\left( {3 - 2x} \right){\rm{d}}x} \)

* Hướng dẫn giải

\(\begin{array}{l} I = 2\int_0^{\frac{\pi }{2}} {f\left( {\sin x} \right)\cos x{\rm{d}}x} + 3\int_0^1 {f\left( {3 - 2x} \right){\rm{d}}x} \\ {\rm{ = }}2\int_0^{\frac{\pi }{2}} {f\left( {\sin x} \right){\rm{d}}\left( {\sin x} \right)} - \frac{3}{2}\int_0^1 {f\left( {3 - 2x} \right){\rm{d}}\left( {3 - 2x} \right)} \\ {\rm{ = }}2\int_0^1 {f\left( x \right){\rm{d}}x} + \frac{3}{2}\int_1^3 {f\left( x \right){\rm{d}}x} \\ {\rm{ }} = 2\int_0^1 {\left( {5 - x} \right){\rm{d}}x} + \frac{3}{2}\int_1^3 {\left( {{x^2} + 3} \right){\rm{d}}x} \\ {\rm{ }} = 9 + 22 = 31 \end{array}\)