Cho hàm số f(x) liên tục trên \(\mathbb{R}\) và \[\mathop \smallint \limits_1^9 \frac{{f\left( {\sqrt x } \right)}}{{\sqrt x }}{\rm{d}}x = 4,\mathop \smallint \limits_0^{\frac{\pi }{2}} f\left( {\sin x} \right)\cos x{\rm{d}}x = 2\]. Tính tích phân \[I = \mathop \smallint \limits_0^3 f\left( x \right){\rm{d}}x\]

Xét tích phân\[\mathop \smallint \limits_1^9 \frac{{f\left( {\sqrt x } \right)}}{{\sqrt x }}{\rm{d}}x = 4\]

Đặt\[t = \sqrt x \Rightarrow {t^2} = x \Rightarrow 2tdt = dx\]

Đổi cận:\(\left\{ {\begin{array}{*{20}{c}}{x = 1 \Rightarrow t = 1}\\{x = 9 \Rightarrow t = 3}\end{array}} \right.\)

Khi đó ta có: \[\mathop \smallint \limits_1^9 \frac{{f\left( {\sqrt x } \right)}}{{\sqrt x }}{\rm{d}}x = \mathop \smallint \limits_1^3 \frac{{f\left( t \right)2tdt}}{t} = 2\mathop \smallint \limits_1^3 f\left( t \right)dt = 2\mathop \smallint \limits_1^3 f\left( x \right)dx\]

\[ \Rightarrow 2\mathop \smallint \limits_1^3 f\left( x \right)dx = 4 \Leftrightarrow \mathop \smallint \limits_1^3 f\left( x \right)dx = 2\]

Xét tích phân\[\mathop \smallint \limits_0^{\frac{\pi }{2}} f\left( {\sin x} \right)\cos x{\rm{d}}x = 2\]

Đặt\[u = \sin x \Rightarrow du = \cos xdx\]

Đổi cận:\(\left\{ {\begin{array}{*{20}{c}}{x = 0 \Rightarrow u = 0}\\{x = \frac{\pi }{2} \Rightarrow u = 1}\end{array}} \right.\)

Khi đó ta có:\[\mathop \smallint \limits_0^{\frac{\pi }{2}} f\left( {\sin x} \right)\cos x{\rm{d}}x = \mathop \smallint \limits_0^1 f\left( u \right)du = \mathop \smallint \limits_0^1 f\left( x \right)dx = 2\]

Vậy \[I = \mathop \smallint \limits_0^3 f\left( x \right){\rm{d}}x = \mathop \smallint \limits_0^1 f\left( x \right){\rm{d}}x + \mathop \smallint \limits_1^3 f\left( x \right){\rm{d}}x = 2 + 2 = 4\]

Đáp án cần chọn là: B