∫5 + x2x4dx is equal to from Mathematics Inte

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Integrals

Multiple Choice Questions

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511.
$\int \frac{\sqrt{5 + x^{2}}}{x^{4}} dx$ is equal to

$\frac{1}{15} {(1 + \frac{5}{x^{2}})}^{\frac{3}{2}}$ + C

$\frac{- 1}{15} {(1 + \frac{1}{x^{2}})}^{\frac{3}{2}} + C$

$\frac{- 1}{15} {(1 + \frac{5}{x^{2}})}^{\frac{3}{2}}$

$\frac{1}{15} {(1 + \frac{1}{x^{2}})}^{\frac{3}{2}}$

C.

$\frac{- 1}{15} {(1 + \frac{5}{x^{2}})}^{\frac{3}{2}}$

$Let I = \int \frac{\sqrt{5 + x^{2}}}{x^{4}} dx = \int \frac{x \sqrt{\frac{5}{x^{2}} + 1}}{x^{4}} dx = \int \frac{\sqrt{\frac{5}{x^{2}} + 1}}{x^{3}} dx Put \frac{5}{x^{2}} + 1 = t \Rightarrow - \frac{2 \times 5}{x^{3}} dx = dt \Rightarrow \frac{1}{x^{3}} dx = - \frac{1}{10} ∴ I = - \frac{1}{10} \int \sqrt{t} dt = - \frac{1}{10} . \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + C = - \frac{1}{15} t^{\frac{3}{2}} + C = - \frac{1}{15} {(\frac{5}{x^{2}} + 1)}^{\frac{3}{2}} + C$

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512.

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

$\frac{1}{e} \log (\frac{1 + e}{2})$
$\log (\frac{1 + e}{2})$
$\frac{1}{e} \log (1 + e)$
$\log (\frac{2}{1 + e})$

513.

The value of the integral $\int_{1}^{e} \frac{1 + \log (x)}{3 x} dx$ is equal to

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

514.

The value of the integtral $\int_{0}^{1} \frac{x^{3}}{1 + x^{8}} dx$ is equal to

$\frac{π}{8}$
$\frac{π}{4}$
$\frac{π}{16}$
$\frac{π}{6}$

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515.

The value of $\int_{2}^{4} (\frac{\log (t)}{t}) dt$ is

$\frac{1}{2} {(\log (2))}^{2}$
$\frac{5}{2} {(\log (2))}^{2}$
$\frac{3}{2} {(\log (2))}^{2}$
${(\log (2))}^{2}$

516.

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

$\log |\cos ({xe}^{x})| + C$
$\log |\cot ({xe}^{x})| + C$
$\log |\sec ({xe}^{- x})| + C$
$\log |\sec ({xe}^{x})| + C$

517.

$\int \frac{x^{5}}{\sqrt{1 + x^{3}}} dx$ is equal to

$\frac{2}{9} \sqrt{1 + x^{2}} (x^{3} - 9)$ + C
$\frac{2}{9} \sqrt{x^{3} - 9} (1 + x^{2})$ + C
$\frac{2}{9} \sqrt{1 + x^{3}}$ + C
$\frac{2}{9} \sqrt{1 + x^{2}} (x^{3} - 2)$ + C

518.

$\int \frac{4 e^{x}}{2 e^{x} - 5 e^{- x}} dx$ is equal to

$4 \log |e^{x} - 5| + C$
$\frac{1}{4} \log |e^{2 x} - 5| + C$
$\log |2 e^{2 x} - 5 e^{- x}| + C$
$\log |2 e^{2 x} - 5| + C$

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519.

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

$\frac{x^{2}}{2} + 2 x + \log |x| + C$
$\frac{x^{2}}{2} + 2 + \log |x| + C$
$\frac{x^{2}}{2} + x + \log |x| + C$
$\frac{x^{2}}{2} + 2 x + 2 \log |x| + C$

520.

$\int \frac{x^{n - 1}}{x^{2 n} + a^{2}} dx$ is equal to

$\frac{1}{na} \tan^{- 1} (\frac{x^{n}}{a}) + C$
$\frac{n}{a} \sin^{- 1} (\frac{x^{n}}{a}) + C$
$\frac{n}{a} \sin^{- 1} (\frac{x^{n}}{a}) + C$
$\frac{n}{a} \cot^{- 1} (\frac{x^{n}}{a}) + C$

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