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Integrals

Multiple Choice Questions

981.

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

$\frac{e^{x}}{x + 1} + C$
$\frac{- e^{x}}{x + 1} + C$
$e^{x} (\frac{x - 1}{x + 1}) + C$
$e^{x} (\frac{x + 1}{x - 1}) + C$

982.

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

$\log |\frac{1 + {xe}^{x}}{{xe}^{x}}| + C$
$\log |\frac{{xe}^{x}}{1 + {xe}^{x}}| + C$
$\log |{xe}^{x} (1 + {xe}^{x})| + C$
$\log |1 + {xe}^{x}| + C$

983.

$\int \frac{f (x) g' (x) - f' (x) g (x)}{f (x) g (x)} [logg (x) - \log (f (x))] dx$ is equal to

$\log [\frac{g (x)}{f (x)}] + C$
$\frac{1}{2} {[\log (\frac{g (x)}{f (x)})]}^{2} + C$
$\frac{g (x)}{f (x)} [\log (\frac{g (x)}{f (x)})] + C$
$[\log (\frac{g (x)}{f (x)})] - \frac{g (x)}{f (x)} + C$

984.

$\int_{0}^{\frac{π}{4}} \frac{\sin (x) + \cos (x)}{3 + \sin (2 x)} dx = ?$

$\frac{1}{2} \log (3)$
$\log (2)$
log(3)
$\frac{1}{4} \log (3)$

985.

$\int_{- 1}^{1} \frac{\sqrt{1 + x + x^{2}} - \sqrt{1 - x + x^{2}}}{\sqrt{1 + x + x^{2}} + \sqrt{1 - x + x^{2}}} dx = ?$

$\frac{3 π}{2}$
$\frac{π}{2}$
0
- 1

986.

$If \int x^{3} e^{5 x} dx = \frac{e^{5 x}}{5^{4}} [f (x)] + C, then f (x) = ?$

$\frac{x^{3}}{5} - \frac{3 x^{2}}{5^{2}} + \frac{6 x}{5^{3}} - \frac{6}{5^{4}}$
$5 x^{3} - 5^{2} x^{2} + 5^{3} x - 6$
$5^{3} x^{3} - 15 x^{2} + 30 x - 6$
$5^{3} x^{3} - 75 x^{2} + 30 x - 6$

987.

$\int \frac{x}{{(x^{2} + 2 x + 2)}^{2}} dx = ?$

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

988.
$If \int \log (a^{2} + x^{2}) dx = h (x) + C, then h (x) = ?$

$xlog (a^{2} + x^{2}) + 2 \tan^{- 1} (\frac{x}{a})$

$x^{2} \log (a^{2} + x^{2}) + x + {atan}^{- 1} (\frac{x}{a})$

$xlog (a^{2} + x^{2}) - 2 x + 2 {atan}^{- 1} (\frac{x}{a})$

$x^{2} \log (a^{2} + x^{2}) + 2 x - a^{2} \tan^{- 1} (\frac{x}{a})$

$xlog (a^{2} + x^{2}) - 2 x + 2 {atan}^{- 1} (\frac{x}{a})$

$Let I = \log (a^{2} + x^{2}) dx By using integration by parts, we get = \log (a^{2} + x^{2}) \int dx - \int \{\frac{d \log (a^{2} + x^{2})}{dx} \int dx\} dx + C = xlog (a^{2} + x^{2}) - \int \frac{2 x}{x^{2} + a^{2}} dx + C = xlog (a^{2} + x^{2}) - 2 x + \frac{2 a^{2}}{a} \tan^{- 1} (\frac{x}{a}) + C = xlog (a^{2} + x^{2}) - 2 x + 2 {atan}^{- 1} (\frac{x}{a}) + C ∴ h (x) = xlog (a^{2} + x^{2}) - 2 x + 2 {atan}^{- 1} (\frac{x}{a})$

989.

$For x > 0, if \int (\log {(x)}^{5}) dx = ? x [A (\log {(x)}^{5}) + B {(\log (x))}^{4} + C {(\log (x))}^{3} + D {(\log (x))}^{2} + E (\log (x)) + F] + Constant, then A + B + C + D + E + F = ?$

- 44
- 42
- 40
- 36

990.

By the definition of the definite integral, the value of $\lim_{n \to \infty} (\frac{1}{\sqrt{n^{2} - 1}} + \frac{1}{\sqrt{n^{2} - 2^{2}}} + . . . \frac{1}{\sqrt{n^{2} - {(n - 1)}^{2}}})$ is equal to

$π$
$\frac{π}{2}$
$\frac{π}{4}$
$\frac{π}{6}$

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Multiple Choice Questions

988. If ∫loga2 + x2dx = hx + C, then hx = ? xloga2 + x2 + 2tan-1xa x2loga2 + x2 + x + atan-1xa xloga2 + x2 - 2x + 2atan-1xa x2loga2 + x2 + 2x - a2tan-1xa