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Integrals

Multiple Choice Questions

371.

$\int \frac{1}{16 x^{2} + 9} dx$ is equal to

$\frac{1}{3} \tan^{- 1} (\frac{4 x}{3}) + c$
$\frac{1}{4} \tan^{- 1} (\frac{4 x}{3}) + c$
$\frac{1}{12} \tan^{- 1} (\frac{4 x}{3}) + c$
$\frac{1}{12} \tan^{- 1} (\frac{3 x}{4}) + c$

372.

The value of $\int_{4}^{7} \frac{{(11 - x)}^{2}}{x^{2} + {(11 - x)}^{2}} dx$ is

373.

$\int (\sqrt{\tan (x)} + \sqrt{\cot (x)}) dx$

$\sqrt{2} \tan^{- 1} (\frac{\tan (x)}{\sqrt{\tan (x)}}) + C$
$\sqrt{2} \tan^{- 1} (\frac{\tan (x) - 1}{\sqrt{2 \tan (x)}}) + C$
$\frac{\tan (x)}{\sqrt{2}} . \tan^{- 1} (\frac{\cot (x) + 1}{\sqrt{2 \tan (x)}}) + C$
$\frac{\tan (x)}{\sqrt{2}} . \tan^{- 1} (\frac{\cot (x) + 1}{\sqrt{2 \tan (x)}}) + C$

374.

$\int \frac{x^{2}}{{(xsin (x) + \cos (x))}^{2}} dx$ is equal to

$\frac{\sin (x) + \cos (x)}{xsin (x) + \cos (x)} + C$
$\frac{x \sin (x) - \cos (x)}{xsin (x) + \cos (x)} + C$
$\frac{\sin (x) - xcos (x)}{xsin (x) + \cos (x)} + C$
None of these

375.

$\int_{0}^{x} \frac{xdx}{1 + \cos (α) \sin (x)}, (0 < α < π)$ is equal to

$\frac{πα}{\sin (α)}$
$\frac{πα}{\cos (α)}$
$\frac{πα}{1 + \sin (α)}$
$\frac{πα}{1 - \cos (α)}$

376.

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

1
0
- 1
None of these

377.

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

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

378.

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

379.

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

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

380.
The value of $\int_{3}^{4} \sqrt{(4 - x) (x - 3)} dx$ is

$\frac{π}{16}$

$\frac{π}{8}$

$\frac{π}{4}$

$\frac{π}{2}$

$\frac{π}{8}$

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

Previous Year Papers

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Multiple Choice Questions

380. The value of ∫344 - xx - 3dx is π16 π8 π4 π2

380.
The value of $\int_{3}^{4} \sqrt{(4 - x) (x - 3)} dx$ is

$\frac{π}{16}$

$\frac{π}{8}$

$\frac{π}{4}$

$\frac{π}{2}$