Tính T=a+b/2+c/3 biết tích phân 0 đến 3.e^(sqrt(1+3x))dx=(a/5)e^2+(b/2)e+c

Đặt \(t = \sqrt {1 + 3x} \Rightarrow {t^2} = 1 + 3x \Rightarrow 2tdt = 3dx \Rightarrow \left\{ {\begin{array}{*{20}{c}} {x = 0,t = 1}\\ {x = 1,t = 2} \end{array}} \right.\) 

\(\Rightarrow \int\limits_0^1 {3{e^{\sqrt {1 + 3x} }}dx} = I = 2\int\limits_1^2 {t.{e^t}} dt\)

Đặt  \(\left\{ {\begin{array}{*{20}{c}} {u = t}\\ {dv = {e^t}dt} \end{array} \Rightarrow \left\{ {\begin{array}{*{20}{c}} {du = dt}\\ {v = {e^t}} \end{array}} \right.} \right.\)

\(\begin{array}{l} \Rightarrow I = 2t.{e^t}\left| {\begin{array}{*{20}{c}} 2\\ 1 \end{array}} \right. - 2\int\limits_1^2 {{e^t}dt} = 2t.{e^t}\left| {\begin{array}{*{20}{c}} 2\\ 1 \end{array} - 2{e^t}\left| {\begin{array}{*{20}{c}} 2\\ 1 \end{array}} \right.} \right. = 2{e^2}\\ \Rightarrow \left\{ {\begin{array}{*{20}{c}} {a = 10}\\ {b = 0}\\ {c = 0} \end{array}} \right. \Rightarrow T = 10 \end{array}\)