First slide

Evaluation of definite integrals

Question

Whenever $a < b,$ the value of $\int_{a}^{b} \frac{| x |}{x} d x$ is

Moderate

A

$b - a$
B

$a - b$
C

$| b | - | a |$
D

$| b | + | a |$

Solution

If $0 \leq a < b,$ then $f (x) = \frac{| x |}{x} = 1$ therefore,
$\int_{a}^{b} f (x) d x = b - a .$ if $a < b \leq 0$ then $f (x) = - 1$ and so
$\int_{a}^{b} f (x) d x = a - b,$ Finally if $a < 0 < b$ then $\int_{a}^{b} f (x) d x =$
$\int_{a}^{0} f (x) d x + \int_{0}^{b} f (x) d x = - (0 - a) + (b - 0) = b - (- a)$
The above three cases can be represented by
$\int_{a}^{b} \frac{| x |}{x} d x = | b | - | a | .$

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