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Integrals

Multiple Choice Questions

491.

$\int_{0}^{10} \frac{x^{10}}{{(10 - x)}^{10} + x^{10}} dx$ is equal to

10
5
2
$\frac{1}{2}$

492.

The value of $\int_{0}^{1} \sqrt{x} e^{\sqrt{x}} dx$ is equal to

$\frac{(e - 2)}{2}$
2(e - 2)
2e - 1
2(e - 1)

493.

$\int \frac{1}{x (\log (x)) \log (\log (x))} dx$ is equal to

$\log (\log (x)) + C$
$\log |\log (xlog (x))| + C$
$\log |\log |\log (\log (x))|| + C$
$\log |\log (\log (x))| + C$

494.

$\int \frac{3^{x}}{\sqrt{1 - 9^{x}}}$ dx is equal to

$\log (3) \sin^{- 1} (3^{x}) + C$
$\frac{1}{3} \sin^{- 1} (3^{x}) + C$
$(\frac{1}{3}) \sin^{- 1} (3^{x}) + C$
$\frac{1}{9} \sin^{- 1} (3^{x}) + C$

495.

The value of the integral $\int \frac{\cos (x)}{\sin (x) + \cos (x)} dx$ is equal to

$x + \log |\sin (x) + \cos (x)| + C$
$\frac{1}{2} [x + \log |\sin (x) + \cos (x)|] + C$
$\log |\sin (x) + \cos (x)| + C$
$\frac{x}{2} + \log |\sin (x) + \cos (x)| + C$

496.

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

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

497.

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

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

498.

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

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

499.
$\int \frac{\sin (x) + \cos (x)}{e^{- x} + \sin (x)} dx$ is equal to

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

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

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

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

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

$Let I = \int \frac{\sin (x) + \cos (x)}{e^{- x} + \sin (x)} dx = \int \frac{e^{x} (\sin (x) + \cos (x))}{(1 + e^{x} \sin (x))} dx Put t = 1 + e^{x} \sin (x) \Rightarrow dt = (e^{x} \sin (x) + e^{x} \cos (x)) dx = e^{x} (\sin (x) + \cos (x)) dx Then, I = \int \frac{dt}{t} = \log |t| + C \Rightarrow I = \log |1 + e^{x} \sin (x)| + C$