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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

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

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

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

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

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

$Let I = \int \frac{4 dx}{x^{2} \sqrt{4 - 9 x^{2}}} = 4 \int \frac{dx}{x^{2} \sqrt{{(2)}^{2} - {(3 x)}^{2}}} Put, 3 x = 2 \sin (θ) \Rightarrow 3 dx = 2 \cos (θ) dθ Then, I = 4 \times \frac{2}{3} \int \frac{\cos (θ)}{\frac{4}{9} \sin^{2} (θ) \sqrt{4 - 4 \sin^{2} (θ)}} = 6 \int \frac{\cos (θ) dθ}{\sin^{2} (θ) . 2 \cos (θ)} = 3 \int \csc^{2} (θ) dθ = - 3 \cot (θ) + C = = - 3 \frac{\sqrt{1 - \sin^{2} (θ)}}{\sin (θ)} + C = \frac{- 3 \sqrt{1 - {(\frac{3 x}{2})}^{2}}}{(\frac{3 x}{2})} + C = - \frac{2}{x} \frac{\sqrt{4 - 9 x^{2}}}{2} + C = - \frac{\sqrt{4 - 9 x^{2}}}{x} + C$