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Integrals

Multiple Choice Questions

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

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

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

0
$2 (\sqrt{2} - 1)$
$2 \sqrt{2}$
$2 (\sqrt{2} + 1)$

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

e + 1
e - 1
e + 2
e

$What is \int \ln (x^{2}) dx = ?$

$2 xln (x) - 2 x + c$
$\frac{2}{x} + c$
$2 xln (x) + c$
$\frac{2 \ln (x)}{x} - 2 x + c$

$What is \int_{0}^{π} e^{x} \sin (x) d x = ?$

$\frac{e^{π} + 1}{2}$
$\frac{e^{π} - 1}{2}$
$e^{π} + 1$
$\frac{e^{π} + 1}{4}$

6.
$What is \int_{1}^{e} xln (x) d x = ?$

$\frac{e + 1}{4}$

$\frac{e^{2} + 1}{2}$

$\frac{e - 1}{2}$

$\frac{e^{2} - 1}{4}$

$\frac{e^{2} + 1}{2}$

$\int_{1}^{e} x . \ln (x) d x = {[\ln (x) . \frac{x^{2}}{2}]}_{1}^{e} - \int_{1}^{e} [\frac{d}{dx} . \ln (x) . \int x . dx] dx = [\ln (e) (\frac{e^{2}}{2}) - \ln (1) x \frac{1}{2}] - \int_{1}^{e} \frac{1}{x} \times \frac{x^{2}}{2} . dx = [1 . \frac{e^{2}}{2} - 0] - \frac{1}{2} {[\frac{x^{2}}{2}]}_{1}^{e} = \frac{e^{2}}{2} - \frac{1}{2} [\frac{e^{2}}{2} - \frac{1}{2}] = \frac{e^{2}}{2} - \frac{e^{2}}{4} + \frac{1}{4} = \frac{e^{2}}{2} + \frac{1}{4} = \frac{e^{2} + 1}{4}$