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211.
$\int {(27 e^{9 x} + e^{12 x})}^{\frac{1}{3}} dx$ is equal to

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

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

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

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

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

$Let I = \int {(27 e^{9 x} + e^{12 x})}^{\frac{1}{3}} dx = \int {(e^{9 x})}^{\frac{1}{3}} {\{27 + e^{3 x}\}}^{\frac{1}{3}} dx = \int e^{3 x} {(27 + e^{3 x})}^{\frac{1}{3}} dx Put t = 27 + e^{3 x} \Rightarrow dt = 3 e^{3 x} dx \Rightarrow I = \frac{1}{3} \int t^{\frac{1}{3}} dt = \frac{1}{3} [t^{\frac{4}{3}}] \times \frac{3}{4} + C \Rightarrow I = \frac{1}{4} {(27 + e^{3 x})}^{\frac{4}{3}} + C$

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

$\sqrt{12}$
$\sqrt{21}$
$\sqrt{19}$
$\sqrt{17}$

217.

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

$\frac{1}{2} \log (2)$
$\log (2)$
$\frac{1}{2} \log (3)$
$\log (3)$

218.

$\int_{0}^{\frac{π}{2}} \frac{dx}{1 + \tan^{3} (x)}$ is equal to

1
$π$
$\frac{π}{2}$
$\frac{π}{4}$

219.

If f(x) = $\int_{1}^{x} \sin^{2} (\frac{t}{2}) dt$ , then the value of $\lim_{x \to 0} \frac{f (π + x) - f (π)}{x}$ is equal to

$\frac{1}{4}$
$\frac{1}{2}$
$\frac{3}{4}$
1

220.

$\int \frac{1}{x^{2} {(x^{4} + 1)}^{\frac{3}{4}}} dx$ is equal to

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

Previous Year Papers

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Multiple Choice Questions

211.
$\int {(27 e^{9 x} + e^{12 x})}^{\frac{1}{3}} dx$ is equal to

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

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

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

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

Previous Year Papers

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Multiple Choice Questions

211. ∫27e9x + e12x13dx is equal to 1427 + e3x13 + C 1427 + e3x23 + C 1327 + e3x43 + C 1427 + e3x43 + C

211.
$\int {(27 e^{9 x} + e^{12 x})}^{\frac{1}{3}} dx$ is equal to

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

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

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

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