Tìm nguyên hàm của hàm số sau: ∫1+sinx1+cosxexdx

Question

\(\int \frac{1+\sin x}{1+\cos x} {\rm e}^{x} {\rm d}x\)

Answer 1

\(I=\int \frac{1+\sin x}{1+\cos x} {\rm e}^{x} {\rm d}x =\int \frac{{\rm e}^{x} }{1+\cos x} {\rm d}x+\int \frac{\sin x.{\rm e}^{x} }{1+\cos x} {\rm d}x   =\int \frac{{\rm e}^{x} }{2\cos ^{2} \frac{x}{2} } {\rm d}x+\int \tan \frac{x}{2} {\rm e}^{x} {\rm d}x  \).

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

\(Vậy: \int \frac{1+\sin x}{1+\cos x} {\rm e}^{x} {\rm d}x =e^{x} \tan \frac{x}{2} +C\)