[Mai Tuấn Long]

Nguyên văn bởi Kratos

Bài 2. Tìm nguyên hàm: $\int{\dfrac{1+\sin x}{1+\cos x}e^xdx} $

Ta có:$\dfrac{1+\sin x}{1+\cos x}$ $=\dfrac{(\tan\frac{x}{2}+1)^2}{2}$

$I=\int\dfrac{1+\sin x}{1+\cos x}e^xdx$ $=\dfrac{1}{2}\int e^x(\tan\frac{x}{2}+1)^2dx$ $=\dfrac{1}{2}\int e^x(\tan^2\frac{x}{2}+1)dx$ $+\int e^x\tan\frac{x}{2}dx$

$=\int e^xd(\tan\frac{x}{2})$ $+\int e^x\tan\frac{x}{2}dx$ $=e^x\tan\frac{x}{2}-\int e^x\tan\frac{x}{2}dx$ $+\int e^x\tan\frac{x}{2}dx$ $=e^x\tan\frac{x}{2}+C$
[RIGHT][I][B]Nguồn: MathScope.ORG[/B][/I][/RIGHT]