(B) : \(I=\int \frac{e^{x}(x-1)}{(x+1)^{3}} d x\) \(=\int \frac{e^{x}(x+1-2)}{(x+1)^{3}} d x\) \(=\int e^{x}\left(\frac{1}{(x+1)^{2}}+\frac{-2}{(x+1)^{3}}\right) d x\) \(=\frac{e^{x}}{(x+1)^{2}}+C\left[\because \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+C\right]\)
(B) : \(\mathrm{I}=\int(1+\mathrm{x}) \log \mathrm{x} d \mathrm{x}\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int \frac{1}{x}\left(x+\frac{x^{2}}{2}\right) d x+C\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int\left(1+\frac{x}{2}\right) d x+C\) \(=\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{2}\right) \cdot \log \mathrm{x}-\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{4}\right)+\mathrm{C}\)
(B) : \(I=\int \frac{e^{x}(x-1)}{(x+1)^{3}} d x\) \(=\int \frac{e^{x}(x+1-2)}{(x+1)^{3}} d x\) \(=\int e^{x}\left(\frac{1}{(x+1)^{2}}+\frac{-2}{(x+1)^{3}}\right) d x\) \(=\frac{e^{x}}{(x+1)^{2}}+C\left[\because \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+C\right]\)
(B) : \(\mathrm{I}=\int(1+\mathrm{x}) \log \mathrm{x} d \mathrm{x}\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int \frac{1}{x}\left(x+\frac{x^{2}}{2}\right) d x+C\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int\left(1+\frac{x}{2}\right) d x+C\) \(=\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{2}\right) \cdot \log \mathrm{x}-\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{4}\right)+\mathrm{C}\)
(B) : \(I=\int \frac{e^{x}(x-1)}{(x+1)^{3}} d x\) \(=\int \frac{e^{x}(x+1-2)}{(x+1)^{3}} d x\) \(=\int e^{x}\left(\frac{1}{(x+1)^{2}}+\frac{-2}{(x+1)^{3}}\right) d x\) \(=\frac{e^{x}}{(x+1)^{2}}+C\left[\because \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+C\right]\)
(B) : \(\mathrm{I}=\int(1+\mathrm{x}) \log \mathrm{x} d \mathrm{x}\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int \frac{1}{x}\left(x+\frac{x^{2}}{2}\right) d x+C\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int\left(1+\frac{x}{2}\right) d x+C\) \(=\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{2}\right) \cdot \log \mathrm{x}-\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{4}\right)+\mathrm{C}\)
(B) : \(I=\int \frac{e^{x}(x-1)}{(x+1)^{3}} d x\) \(=\int \frac{e^{x}(x+1-2)}{(x+1)^{3}} d x\) \(=\int e^{x}\left(\frac{1}{(x+1)^{2}}+\frac{-2}{(x+1)^{3}}\right) d x\) \(=\frac{e^{x}}{(x+1)^{2}}+C\left[\because \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+C\right]\)
(B) : \(\mathrm{I}=\int(1+\mathrm{x}) \log \mathrm{x} d \mathrm{x}\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int \frac{1}{x}\left(x+\frac{x^{2}}{2}\right) d x+C\) \(=\log x \cdot\left(x+\frac{x^{2}}{2}\right)-\int\left(1+\frac{x}{2}\right) d x+C\) \(=\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{2}\right) \cdot \log \mathrm{x}-\left(\mathrm{x}+\frac{\mathrm{x}^{2}}{4}\right)+\mathrm{C}\)
