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Integrals

Multiple Choice Questions

61.

The value of integral $\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{(\sin (x) - xcos (x))}{x (x + \sin (x))} d x$

$\log_{e} (\frac{2 (π + 3)}{(2 π + 3 \sqrt{3})})$
$\log_{e} (\frac{π + 3}{2 (2 π + 3 \sqrt{3})})$
$\log_{e} (\frac{2 π + 3 \sqrt{3}}{2 (π + 3)})$
$\log_{e} (\frac{2 (2 π + 3 \sqrt{3})}{π + 3})$

62.

If F(x) = $\int_{0}^{x} \frac{\cos (t)}{(1 + t^{2})} d t, 0 \leq x \leq 2 π$ . Then,

F is decreasing in $(\frac{π}{2}, \frac{3 π}{2})$ and decreasing in $(0, \frac{π}{2}) and (\frac{3 π}{2}, 2 π)$
F is increasing in (0, $π$ ) and decreasing in ( $π, 2 π$ ).
Fis increasing in ( $π, 2 π$ ) and decreasing in ( $0, π$ ).
Fis increasing in (0, $\frac{π}{2}$ ) and $(\frac{3 π}{2}, 2 π)$ and decreasing in ( $\frac{π}{2}, \frac{3 π}{2}$ ).

63.

The value of the integral $\int_{\frac{π}{6}}^{\frac{π}{2}} (\frac{1 + \sin (2 x) + \cos (2 x)}{\sin (x) + \cos (x)}) d x$ is equal to

64.
The value of the integral $\int_{0}^{\frac{π}{2}} \frac{1}{1 + {(\tan (x))}^{101}} d x$ is equal to

1

$\frac{π}{6}$

$\frac{π}{8}$

$\frac{π}{4}$

$\frac{π}{4}$

$Let I = \int_{0}^{\frac{π}{2}} \frac{1}{1 + {(\tan (x))}^{101}} d x = \int_{0}^{\frac{π}{2}} \frac{d x}{1 + {\{\tan (\frac{π}{2} - x)\}}^{101}} = \int_{0}^{\frac{π}{2}} \frac{d x}{1 + {(\cot (x))}^{101}} = \int_{0}^{\frac{π}{2}} \frac{\tan {(x)}^{101}}{\tan {(x)}^{101} + 1} d x = \int_{0}^{\frac{π}{2}} \frac{1 + \tan {(x)}^{101} - 1}{1 + \tan {(x)}^{101}} d x = {[x]}_{0}^{\frac{π}{2}} - I$