Cho hàm số y=f(x) liên tục trên

* Hướng dẫn giải

Đáp án A

Ta có: $f\left(x\right)+f^{\text{'}}\left(x\right)=\frac{\sqrt{4\text{x}+1}}{e^x}\iff f^{\text{'}}\left(x\right).e^x+f\left(x\right).e^x=\sqrt{4\text{x}+1}$

$\iff {\left[f\left(x\right)e^x\right]}^{\text{'}}=\sqrt{4\text{x}+1}\iff \underset{0}{\overset{2}{\int }}{\left[f\left(x\right)e^x\right]}^{\text{'}}d\text{x}=\underset{0}{\overset{2}{\int }}\sqrt{4\text{x}+1}d\text{x}\iff {\left.f\left(x\right)e^x\right|}_0^2=\frac{13}{3}$

$\iff f\left(2\right).e^2-f\left(0\right)=\frac{13}{3}\iff f\left(2\right)=\frac{16}{3{\text{e}}^2}\approx 0,72\in (0;1)$.