Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

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

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

$\frac{π}{16}$

$\frac{π}{8}$

$\frac{π}{4}$

$\frac{π}{2}$

$\frac{π}{8}$

$Let I = \int_{3}^{4} \sqrt{(4 - x) (x - 3)} dx = \int_{3}^{4} \sqrt{- x^{2} + 7 x - 12} dx = \int_{3}^{4} \sqrt{- (x^{2} - 7 x + \frac{49}{4} - \frac{49}{4}) - 12} dx = \int_{3}^{4} \sqrt{- {(x - \frac{7}{2})}^{2} + \frac{49}{4} - 12} dx = \int_{3}^{4} \sqrt{\frac{1}{4} - {(x - \frac{7}{2})}^{2}} dx Let t = x - \frac{7}{2} \Rightarrow dt = dx ∴ Upper limit = \frac{1}{2} and lower limit = - \frac{1}{2} = \int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{{(\frac{1}{2})}^{2} - t^{2}} dt = 2 \int_{0}^{\frac{1}{2}} \sqrt{{(\frac{1}{2})}^{2} - t^{2}} dt = 2 {[\frac{t}{2} \sqrt{\frac{1}{4} - t^{2}} + \frac{1}{8} \sin^{- 1} (2 t)]}_{0}^{\frac{1}{2}} = 2 [0 + \frac{1}{8} \times \frac{π}{2}] = \frac{π}{8}$

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