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Integrals

Multiple Choice Questions

661.
$\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos (x)}{1 + e^{x}} d x$

1

0

- 1

None of these

$I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos (x)}{1 + e^{x}} d x . . . (i) I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos (\frac{π}{2} - \frac{π}{2} - x)}{1 + e^{(\frac{π}{2} - \frac{π}{2} - x)}} dx = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos (- x) dx}{1 + e^{- x}} I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos (x)}{1 + e^{x}} dx . . . (ii) = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{e^{x} \cos (x)}{1 + e^{x}} dx On adding Eqs . (i) and (ii), we get 2 I = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{(1 + e^{x}) \cos (x)}{(1 + e^{x})} = \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (x) dx = \int_{0}^{\frac{π}{2}} \cos (x) dx [∵ Since, \cos (x) is an even function] ∴ 2 I = 2 {[\sin (x)]}_{0}^{\frac{π}{2}} = 2 (1 - 0) = 2 \Rightarrow I = 1$

662.

$\int_{0}^{\frac{π}{2}} \frac{dx}{1 + \tan (x)}$ is equal to

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

663.

By Simpson rule taking n = 4, the value of the integral $\int_{0}^{1} \frac{1}{1 + x^{2}} dx$ is equal to

0.788
0.781
0.785
None of the above

664.

The value of $\int_{0}^{π} \log (1 + \cos (x)) dx$ is

$- \frac{π}{2} \log (2)$
$πlog (\frac{1}{2})$
$πlog (2)$
$\frac{π}{2} \log (2)$

665.

The value of $\int_{3}^{4} \sqrt{(4 - x) (x - 3)} dx$ is

$\frac{π}{16}$
$\frac{π}{8}$
$\frac{π}{4}$
$\frac{π}{2}$

666.

The value of $\int \frac{dx}{x (x^{n} + 1)}$ is

$\frac{1}{n} \log (\frac{x^{n}}{x^{n} + 1}) + C$
$\log (\frac{x^{n} + 1}{x^{n}}) + C$
$\frac{1}{n} \log (\frac{x^{n} + 1}{x^{n}}) + C$
$\log (\frac{x^{n}}{x^{n} + 1}) + C$

667.

The value of $\int \cos (\log (x)) dx$ is

$\frac{1}{2} \{\sin (\log (x)) + \cos (\log (x))\} + C$
$\frac{x}{2} \{\sin (\log (x)) + \cos (\log (x))\} + C$
$\frac{x}{2} \{\sin (\log (x)) - \cos (\log (x))\} + C$
$\frac{1}{2} \{\sin (\log (x)) - \cos (\log (x))\} + C$

668.

The value of $\int e^{x} [\frac{1 + \sin (x)}{1 + \cos (x)}] dx$ is

$\frac{1}{2} e^{x} \sec (\frac{x}{2}) + C$
$e^{x} \sec (\frac{x}{2}) + C$
$\frac{1}{2} e^{x} \tan (\frac{x}{2}) + C$
$e^{x} \tan (\frac{x}{2}) + C$

669.

The value of $\int \frac{1}{3 \sin (x) - \cos (x) + 3} dx$ is

$\log (\frac{\tan (\frac{x}{2}) + 1}{2 \tan (\frac{x}{2}) + 1})$ + C
$\frac{1}{2} \log (\frac{2 \tan (\frac{x}{2}) + 1}{\tan (\frac{x}{2}) + 1})$ + C
$\log (\frac{2 \tan (\frac{x}{2}) + 1}{\tan (\frac{x}{2}) + 1})$ + C
$2 \log (\frac{2 \tan (\frac{x}{2}) + 1}{\tan (\frac{x}{2}) + 1})$ + C