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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.

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

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

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

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

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

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

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

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

$Let I = \int \frac{5 x + 3}{x^{2} (x^{2} - 2)} dx Now, consider \frac{5 x + 3}{x^{2} (x^{2} - 2)} and take x^{2} = y Then, \frac{5 x + 3}{x^{2} (x^{2} - 2)} = \frac{5 y + 3}{y (y - 2)} (i) Now, put \frac{5 y + 3}{y (y - 2)} = \frac{A}{y} + \frac{B}{y - 2} (ii) \Rightarrow 5 y + 3 = (y - 2) A + yB \Rightarrow 5 y + 3 = y (A + B) - 2 A On compairing, we get A + B = 5 and 3 = - 2 A \Rightarrow A = \frac{- 3}{2} and B = \frac{13}{2} Thus, \frac{5 x^{2} + 3}{x^{2} (x^{2} - 2)} = - \frac{3}{2} \frac{1}{x^{2}} + \frac{13}{2} \frac{1}{x^{2} - 2} [Using Eqs (i) and (ii)] Hence, I = - \frac{3}{2} \int \frac{dx}{x^{2}} + \frac{13}{2} \int \frac{dx}{x^{2} - 2} = - \frac{3}{2} (- \frac{1}{x}) + \frac{13}{2} \times \frac{1}{2 \sqrt{2}} \log (|\frac{x - \sqrt{2}}{x + \sqrt{2}}|) + C = \frac{3}{2 x} + \frac{13}{4 \sqrt{2}} \log (|\frac{x - \sqrt{2}}{x + \sqrt{2}}|) + C$