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Integrals

Multiple Choice Questions

341.

Suppose $M = \int_{0}^{π / 2} \frac{\cos (x)}{x + 2} d x, N = \int_{0}^{π / 4} \frac{\sin (x) \cos (x)}{{(x + 1)}^{2}} d x$ . Then, the values of (M - N) equals

$\frac{3}{π + 2}$
$\frac{2}{π - 4}$
$\frac{4}{π - 2}$
$\frac{2}{π + 4}$

342.

The value of the integral

$\int_{- 1}^{1} \{\frac{x^{2013}}{e^{|x|} (x^{2} + \cos (x))} + \frac{1}{e^{|x|}}\} d x$ is equal to

0
$1 - e^{- 1}$
$2 e^{- 1}$
$2 (1 - e^{- 1})$

343.

The value of I = $\int_{0}^{π / 4} (\tan^{n + 1} (x)) d x + \frac{1}{2} \int_{0}^{π / 2} (\tan^{n + 1} (\frac{x}{2})) d x$ is

$\frac{1}{n}$
$\frac{n + 2}{2 n + 1}$
$\frac{2 n - 1}{n}$
$\frac{2 n - 3}{3 n - 2}$

344.

The value of the integral

$\int_{1}^{2} e^{x} (\log_{e} (x) + \frac{x + 1}{x}) d x$

$e^{2} (1 + \log_{e} (2))$
$e^{2} - e$
$e^{2} (1 + \log_{e} (2)) - e$
$e^{2} - e (1 + \log_{e} (2))$

345.

If [a] denote the greatest integer which is less than or equal to a. Then, the value of the integral $\int_{- \frac{π}{2}}^{\frac{π}{2}} [\sin (x) \cos (x)] d x$ is

$\frac{π}{2}$
$π$
$- π$
$- \frac{π}{2}$

346.
The value of integral $\int_{\frac{π}{6}}^{\frac{π}{3}} \frac{(\sin (x) - xcos (x))}{x (x + \sin (x))} d x$

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

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

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

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

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

$Let I = \int_{- \frac{π}{6}}^{\frac{π}{3}} \frac{(\sin (x) - xcos (x))}{x (x + \sin (x))} dx$

$\Rightarrow I = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{(x + \sin (x) - x (1 + \cos (x)))}{x (x + \sin (x))} d x$

$\Rightarrow I = \int_{- \frac{π}{6}}^{\frac{π}{3}} (\frac{1}{x} - \frac{1 + \cos (x)}{x + \sin (x)}) d x$

$\Rightarrow I = {[\log (x)]}_{\frac{π}{6}}^{\frac{π}{3}} - \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{1 + \cos (x)}{x + \sin (x)} d x \{Put t = x + \sin (x) in IInd term dt = (1 + \cos (x)) dx\}$

$\Rightarrow I = (\log (\frac{π}{3}) - \log (\frac{π}{6})) - \int_{(\frac{π}{6} + \frac{1}{2})}^{(\frac{π}{3} + \frac{\sqrt{3}}{2})} \frac{d t}{t}$

$\Rightarrow I = \log (2) - {[\log (t)]}_{(\frac{π}{6} + \frac{1}{2})}^{(\frac{π}{3} + \frac{\sqrt{3}}{2})}$

$\Rightarrow I = \log (2) - [\log (\frac{π}{3} + \frac{\sqrt{3}}{2}) - \log (\frac{π}{6} + \frac{1}{2})]$

$\Rightarrow I = \log (2) - \log (\frac{2 π + 3 \sqrt{3}}{π + 3}) (∵ \log (m) - \log (n) = \log (\frac{m}{n}))$

$\Rightarrow I = \log (\frac{2 (π + 3)}{2 π + 3 \sqrt{3}}) \Rightarrow I = \log (\frac{2 π + 6}{2 π + 3 \sqrt{3}})$

347.

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 ( $\frac{π}{2}, \frac{3 π}{2}$ ).