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Integrals

Multiple Choice Questions

211.

$\int {(27 e^{9 x} + e^{12 x})}^{\frac{1}{3}} dx$ is equal to

$(\frac{1}{4}) {(27 + e^{3 x})}^{\frac{1}{3}} + C$
$(\frac{1}{4}) {(27 + e^{3 x})}^{\frac{2}{3}} + C$
$(\frac{1}{3}) {(27 + e^{3 x})}^{\frac{4}{3}} + C$
$(\frac{1}{4}) {(27 + e^{3 x})}^{\frac{4}{3}} + C$

212.

$\int \frac{4 dx}{x^{2} \sqrt{4 - 9 x^{2}}}$ is equal to

$\sqrt{4 - 9 x^{2}} + C$
$\frac{- 2}{3} \sqrt{4 - 9 x^{2}} + C$
$\frac{- \sqrt{4 - 9 x^{2}}}{x} + C$
$\frac{2}{3} \frac{\sqrt{4 - 9 x^{2}}}{x} + C$

213.

$\int e^{- x} \csc (x) (1 + \cot (x)) dx$ is equal to

$e^{- x} \csc (x) + C$
$- e^{- x} \csc (x) + C$
- $e^{- x} (\csc (x) + \cot (x)) + C$
- $e^{- x} (\csc (x) - \tan (x)) + C$

214.

$\int \frac{\sin (x) + \cos (x)}{e^{- x} + \sin (x)} dx$ is equal to

$\log |1 - e^{x} \sin (x)| + C$
$\log |1 + e^{x} \sin (x)| + C$
$\log |1 + e^{x} \sin (x)| + C$
$\log |1 - e^{- x} \sin (x)| + C$

215.
The solution of $\int_{\sqrt{2}}^{x} \frac{dt}{\sqrt{t^{2} - 1}} = \frac{π}{12}$ is

1

2

3

4

$Given, \int_{\sqrt{2}}^{x} \frac{dt}{\sqrt{t^{2} - 1}} = \frac{π}{12} \Rightarrow {[\sec^{- 1} (t)]}_{\sqrt{2}}^{x} = \frac{π}{12} \Rightarrow \sec^{- 1} (x) - \sec^{- 1} (\sqrt{2}) = \frac{π}{12} \Rightarrow \sec^{- 1} (x) = \frac{π}{12} + \frac{π}{4} = \frac{4 π}{12} = \frac{π}{3} ∴ x = \sec (\frac{π}{3}) = 2$