Wiskunde Multiple Choice Vraag

De bepaalde integraal $\int_{0}^{e-1}\frac{x-1}{x+1}\,dx$ is gelijk aan	A. 0
B. e − 1
C. e − 2
D. e − 3
E. e − 4

$\int_{0}^{e-1}{\frac{x-1}{x+1}dx=}\int_{0}^{e-1}{\frac{x+1-2}{x+1}dx=\int_{0}^{e-1}{dx-2\int_{0}^{e-1}{\frac{1}{x+1}dx}=}\left[x\right]_0^{e-1}-2.{ln\left|x+1\right|}_0^{e-1}}$
$\int_{0}^{e-1}{\frac{x-1}{x+1}dx=}\int_{0}^{e-1}{\frac{x+1-2}{x+1}dx=\int_{0}^{e-1}{dx-2\int_{0}^{e-1}{\frac{1}{x+1}dx}=}\left[x\right]_0^{e-1}-2.{ln\left|x+1\right|}_0^{e-1}}$
= e − 1 − 0 − 2( ln |e | − ln |0+1| ) = e − 1 − 2 ( 1 − 0) = e − 1 − 2 = e – 3