First slide

Evaluation of definite integrals

Question

The integral $\int_{- 1 / 2}^{1 / 2} ([x] + 2 \log \frac{1 + x}{1 - x}) d x$ equal ((where [x] is greatest integer function)

Moderate

A

– 1/2
B

0
C

1
D

2 log (1/2)

Solution

$\int_{- 1 / 2}^{1 / 2} ([x] + 2 \log \frac{1 + x}{1 - x}) d x = \int_{- 1 / 2}^{1 / 2} [x] d x$
(since log $\frac{1 + x}{1 - x}$ is an odd function)
$= \int_{- 1 / 2}^{0} [x] d x + \int_{0}^{1 / 2} [x] d x = \int_{- 1 / 2}^{0} (- 1) d x = - \frac{1}{2} .$

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