Tích phân ∫dx/(ex+1) bằng

Question

Tích phân \(\int _{0}^{m}\frac{dx}{e^{x} +1}\)   bằng 

\( A. m-\ln \left(e^{m} +1\right)-\ln 2. \)

\(B. m+\ln \left(e^{m} +1\right)-\ln 2. \)

\( C. m+\ln \left(e^{m} +1\right)+\ln 2. \)

\(D. m-\ln \left(e^{m} +1\right)+\ln 2.\)

Answer 1

Chọn D
\(\int _{0}^{m}\frac{dx}{e^{x} +1}  =\int _{0}^{m}\frac{e^{x} +1-e^{x} }{e^{x} +1}  dx\)

\(=\int _{0}^{m}\left(1-\frac{e^{x} }{e^{x} +1} \right) dx=\int _{0}^{m}dx -\int _{0}^{m}\frac{e^{x} }{e^{x} +1}  dx  \)

\(=\int _{0}^{m}dx -\int _{0}^{m}\frac{d\left(e^{x} +1\right)}{e^{x} +1}  \)

\(=\left. x\right|_{0}^{m} -\left. \ln \left(e^{x} +1\right)\right|_{0}^{m} =m-\ln \left(e^{m} +1\right)+\ln 2. \)