I =∫_a^b |x|x dx case 1 : b gt; #160; a gt; 0 I =∫_a^b|x|x dx = ∫_a^b xx dx = ∫_a^b dx = b a = |b| |a| case 2 : b gt; 0 gt; a ...

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Global Maxima

Evaluate ∫ab...

Question

Evaluate
$\int_{a}^{b} \frac{| x |}{x} d x$

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Solution

$I = \int_{a}^{b} \frac{| x |}{x} d x$

case 1 : $b > a > 0$

$I = \int_{a}^{b} \frac{| x |}{x} d x = \int_{a}^{b} \frac{x}{x} d x = \int_{a}^{b} d x = b - a = | b | - | a |$

case 2 : $b > 0 > a$

$I = \int_{a}^{b} \frac{| x |}{x} d x = \int_{a}^{0} \frac{- x}{x} d x + \int_{0}^{b} \frac{x}{x} d x = \int_{a}^{0} - d x + \int_{0}^{b} d x = | b | - | a |$

Case 3: $0 > b > a$

$I = \int_{a}^{b} \frac{| x |}{x} d x = \int_{a}^{b} - \frac{x}{x} d x = \int_{a}^{b} - d x = - (b - a) = | b | - | a |$

$I = \int_{a}^{b} \frac{| x |}{x} d x = | b | - | a |$ .

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