Tìm nguyên hàm của hàm số sau: ∫(tan2x+tanx+1)exdx

Question

\(\int \left(\tan ^{2} x+\tan x+1\right){\rm e}^{x} {\rm d}x \)

Answer 1

Đặt: \(I_{3} =\int \left(\tan ^{2} x+\tan x+1\right){\rm e}^{x} {\rm d}x =\int \left(\tan ^{2} x+1\right){\rm e}^{x} {\rm d}x +\int \tan x.{\rm e}^{x} {\rm d}x\) .

 Với\(\int \tan x.{\rm e}^{x}  {\rm d}x.\)  

 Đặt\( \left[\begin{array}{l} {u=\tan x} \\ {{\rm d}v={\rm e}^{x} {\rm d}x} \end{array}\right. \Rightarrow \left[\begin{array}{l} {{\rm d}u=\left(\tan ^{2} x+1\right){\rm d}x} \\ {v={\rm e}^{x} } \end{array}\right. .\)
\(\Rightarrow \int \tan x.{\rm e}^{x}  {\rm d}x=\tan x.{\rm e}^{x} -\int \left(\tan ^{2} x+1\right){\rm e}^{x} {\rm d}x +C. \Rightarrow I_{3} =\tan x.e^{x} +C\)