First slide

Evaluation of definite integrals

Question

$\int_{- \frac{1}{2}}^{\frac{1}{2}} ([x] + \ln (\frac{1 + x}{1 - x})) d x =$

Moderate

A

$- \frac{1}{2}$
B

0
C

1
D

$2 \ln \frac{1}{2}$

Solution

Since $\ln (\frac{1 + x}{1 - x})$ is odd,
$I = \int_{- \frac{1}{2}}^{\frac{1}{2}} [x] d x = \int_{- \frac{1}{2}}^{0} (- 1) d x + \int_{0}^{\frac{1}{2}} 0. d x$ $= - \frac{1}{2}$

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