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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

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

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

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

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

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

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

$What is \int_{0}^{\sqrt{2}} [x^{2}] d x equal to (where [.] is the greatest integer function) ?$

$\sqrt{2} - 1$
$1 - \sqrt{2}$
$2 (\sqrt{2} - 1)$
$\sqrt{3} - 1$

$What is \int \frac{dx}{2^{x} - 1} = ?$

$\ln (2^{x} - 1) + c$
$\frac{\ln (1 - 2^{- x})}{\ln (2)} + c$
$\frac{\ln (2^{x} - 1)}{2 \ln (2)} + c$
$\frac{\ln (1 + 2^{- x})}{\ln (2)} + c$

$What is the value of \int_{\frac{- π}{4}}^{\frac{π}{4}} (\sin (x) - \tan (x)) d x ?$

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

10.
$If \int_{a}^{b} x^{3} d x = 0 and \int_{a}^{b} x^{3} d x = \frac{2}{3}, then what are the values of a and b respectivley ?$

- 1, 1

1, 1

0, 0

2, - 2

- 1, 1

$We have, \int_{a}^{b} x^{3} d x = 0 {[\frac{x^{4}}{4}]}_{a}^{b} = 0 \frac{b^{4}}{4} - \frac{a^{4}}{4} = 0 b^{4} - a^{4} = 0 . . . (i) (b - a) (b + a) (b^{2} + a^{2}) = 0 b = a, b = - a, b^{2} \neq a^{2} and \int_{a}^{b} x^{2} d x = \frac{2}{3} [\frac{b^{3}}{3} - \frac{a^{3}}{3}] = \frac{2}{3} b^{3} - a^{3} = 2 . . . (ii) Put value b = - a in the equation {(- a)}^{3} - a^{3} = 2 - 2 a^{3} = 2 a^{3} = - 1 a = - 1 b = 1 Hence, the value of a and b are - 1, 1 respectively .$

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Test Yourself

Multiple Choice Questions

10. If ∫abx3dx = 0 and ∫abx3dx = 23, then what are the values of a and b respectivley ? - 1, 1 1, 1 0, 0 2, - 2

10.
$If \int_{a}^{b} x^{3} d x = 0 and \int_{a}^{b} x^{3} d x = \frac{2}{3}, then what are the values of a and b respectivley ?$

- 1, 1

1, 1

0, 0

2, - 2