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Integrals

Multiple Choice Questions

What is $\int e^{xln (a)} dx = ?$

$\frac{a^{x}}{\ln (a)} + c$
$\frac{e^{x}}{\ln (a)} + c$
$\frac{e^{x}}{\ln (ae)} + c$
$\frac{{ae}^{x}}{\ln (a)} + c$

2.
$\int_{0}^{\frac{π}{2}} |\sin (x) - \cos (x)| d x = ?$

0

$2 (\sqrt{2} - 1)$

$2 \sqrt{2}$

$2 (\sqrt{2} + 1)$

$2 (\sqrt{2} - 1)$

$We have, \int_{0}^{\frac{π}{2}} |\sin (x) - \cos (x)| d x$

$= \int_{0}^{\frac{π}{4}} |\sin (x) - \cos (x)| d x + \int_{\frac{π}{4}}^{\frac{π}{2}} |\sin (x) - \cos (x)| d x = \int_{0}^{\frac{π}{4}} (\cos (x) - \sin (x)) d x + \int_{\frac{π}{4}}^{\frac{π}{2}} (\sin (x) - \cos (x)) d x = {[\sin (x) + \cos (x)]}_{0}^{\frac{π}{4}} + {[- \cos (x) - \sin (x)]}_{\frac{π}{4}}^{\frac{π}{2}} = [(\sin (\frac{π}{4}) + \cos (\frac{π}{4})) - (\sin (0) + \cos (0))] + [(- \cos (\frac{π}{2}) - \sin (\frac{π}{2})) - (- \cos (\frac{π}{4}) - \sin (\frac{π}{4}))]$

$= [(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}) - (0 + 1)] + [(- 0 - 1) - (- \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}})] = [\frac{2}{\sqrt{2}} - 1] + [- 1 - (- \frac{2}{\sqrt{2}})] = [\sqrt{2} - 1] + [- 1 + \sqrt{2}] = 2 \sqrt{2} - 2 = 2 (\sqrt{2} - 1)$