Cho hàm số y= f(x) = 4x khi x>2 -2x + 12 khi x nhỏ hơn hoặc bằng 2

Câu hỏi :

Cho hàm số $y=f\left(x\right)=\left\{\begin{array}{l}4x\text{\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.33em}}x>2\\ -2x+12\text{\hspace{1em}khi\hspace{0.33em}}x\le 2\end{array}\right.$. Tính tích phân $I=\underset{0}{\overset{\sqrt{3}}{\int }}\frac{x.f\left(\sqrt{x^2+1}\right)}{\sqrt{x^2+1}}dx+4\underset{\ln 2}{\overset{\ln 3}{\int }}{e}^{2x}.f\left(1+e^{2x}\right)dx$.