Evaluate: ^ 3/2 _ 0 |x x| dx. - Vidyadip

Question

Evaluate: $\int\limits^{3/2}_{0} |x \sin \pi \text{ } x| \text{dx}.$

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Answer

$\text{I} = \int\limits^{3/2}_{0} | \text{x} \sin \pi \text{ x}| \text{dx}$
$= \int\limits^{1}_{0} \text{x} \sin \pi \text{ x} . \text{dx} - \int\limits^{3/2}_{1} \text{x} \sin \pi \text{x dx}$
$= \bigg[-\text{x} \frac{\cos \pi \text{x}}{\pi} + \frac{\sin \pi\text{x}}{\pi^{2}} \bigg]^{1}_{0} - \bigg[-\frac{\text{x}\cos \pi \text{x}}{\pi} + \frac{\sin \pi \text{x}}{\pi}^{2} \bigg]^{3/2}_{1}$
$= \frac{2}{\pi} + \frac{1}{\pi^{2}}$

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