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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

216.
Let f(x) = x² - 2. If $\int_{3}^{6} f (x) dx = 3 f (c)$ for some c $\in$ (3, 6), then the value of c is equal to

$\sqrt{12}$

$\sqrt{21}$

$\sqrt{19}$

$\sqrt{17}$

$\sqrt{21}$

$Given, f (x) = x^{2} - 2 . . . (i) Now, \int_{3}^{6} f (x) dx = 3 f (c) \Rightarrow \int_{3}^{6} (x^{2} - 2) dx = 3 f (c) [∵ from Eq . (i)] \Rightarrow {[\frac{x^{3}}{3} - 2 x]}_{3}^{6} = 3 f (c) \Rightarrow [72 - 12 - 9 + 6] = 3 f (c) \Rightarrow 78 - 21 = 3 f (c) \Rightarrow 3 f (c) = 57 \Rightarrow f (c) = 19 \Rightarrow c^{2} - 2 = 19 [∵ from Eq . (i)] \Rightarrow c^{2} = 21 \Rightarrow c = \sqrt{21} which lies between (3, 6) .$