Tính tích phân sau: ∫(x(1+sinx)−sinx)/(1−x)dx

Question

Tính tích phân sau:

\(\int _{-\pi }^{0}\frac{x\left(1+\sin x\right)-\sin x}{1-x} {\rm d}x\)

Answer 1

\( \int _{-\pi }^{0}\frac{x\left(1+\sin x\right)-\sin x}{1-x} {\rm d}x                     \)
\(I=\int _{-\pi }^{0}\frac{\left(x-1\right)\sin x+x}{1-x}  {\rm d}x=\int _{-\pi }^{0}-\sin x{\rm d}x+\int _{-\pi }^{0}\frac{x-1+1}{1-x} {\rm d}x  \) 

\(=\int _{-\pi }^{0}-\sin x{\rm d}x +\int _{-\pi }^{0}\left(-1+\frac{1}{1-x} \right){\rm d}x \)
\(=\left(cosx-x-\ln \left|1-x\right|\right)|_{-\pi }^{0}\)

\(=1-\left(-1+\pi -\ln \left(1+\pi \right)\right)=2-\pi +\ln \left(1+\pi \right) \)