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Integrals

Multiple Choice Questions

701.

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

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

702.
$\int \frac{x}{{(x^{2} + 2 x + 2)}^{2}} dx = ?$

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

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

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

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

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

$Let I = \int \frac{x}{{(x^{2} + 2 x + 2)}^{2}} dx I = \int \frac{x}{{[{(x + 1)}^{2} + 1]}^{2}} dx Put x + 1 = \tan (θ) \Rightarrow dx = \sec^{2} (θ) ∴ I = \int \frac{(\tan (θ) - 1) \sec^{2} (θ)}{{(\tan^{2} (θ) + 1)}^{2}} dθ I = \int \frac{(\tan (θ) - 1) \sec^{2} (θ)}{\sec^{4} (θ)} dθ = \int \frac{(\tan (θ) - 1)}{\sec^{2} (θ)} dθ = \int (\frac{\tan (θ)}{\sec^{2} (θ)} - \frac{1}{\sec^{2} (θ)}) dθ = \int (\sin (θ) \cos (θ) - \cos^{2} (θ)) dθ$

$= \frac{1}{2} \int (2 \sin (θ) \cos (θ) - 2 \cos^{2} (θ)) dθ = \frac{1}{2} (\sin (2 θ) - 1 - \cos (2 θ)) dθ = \frac{1}{2} [\frac{- \cos (2 θ)}{2} - θ - \frac{\sin (2 θ)}{2}] + C = - \frac{1}{4} [\cos (2 θ) + \sin (2 θ)] - \frac{1}{2} θ + C = - \frac{1}{4} [\frac{1 - {(x + 1)}^{2} + 2 (x + 1)}{1 + {(x + 1)}^{2}}] - \frac{1}{2} \tan^{- 1} (x + 1) + C = \frac{1}{4} (\frac{x^{2} - 2}{x^{2} + 2 x + 2}) - \frac{1}{2} \tan^{- 1} (x + 1) + C$

703.

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

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

704.

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

- 44
- 42
- 40
- 36

705.

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

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

706.

$\int_{\frac{π}{4}}^{\frac{π}{4}} (\frac{x + \frac{π}{4}}{2 - \cos (2 x)}) dx$ is equal to

$\frac{8 π \sqrt{3}}{5}$
$\frac{2 π \sqrt{3}}{9}$
$\frac{4 π^{2} \sqrt{3}}{9}$
$\frac{π^{2}}{6 \sqrt{3}}$

707.

$\int \frac{5 x + 3}{x^{2} (x^{2} - 2)} dx = ?$

$\frac{3}{2 x} + \frac{\sqrt{13}}{2 \sqrt{2}} \log (|\frac{\sqrt{2} - x}{\sqrt{2} + x}|) + C$
$\frac{3}{2 x} + \frac{13}{4 \sqrt{2}} \log (|\frac{x + \sqrt{2}}{x - \sqrt{2}}|) + C$
$\frac{3}{2 x} + \frac{13}{4 \sqrt{2}} \log (|\frac{x - \sqrt{2}}{x + \sqrt{2}}|) + C$
$\frac{3}{5} x + \frac{5}{3 \sqrt{2}} \log (|\frac{x + \sqrt{2}}{x - \sqrt{2}}|) + C$