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Integrals

Multiple Choice Questions

621.

$\int_{- 1}^{1} ({ax}^{3} + bx) dx = 0$ for

any value of a and b
a > 0, b > 0 only
a > 0, b > 0 only
a < 0, b > 0 only

622.

Using the Trapezoidal rule, the approximate value of $\int_{1}^{4} ydx$

x 1 2 3 4

y 0.7111 0.7222 0.7333 0.7444

0.1833
1.1833
2.1833
3.1833

623.

$\int \frac{dx}{1 - \cos (x) - \sin (x)} is equal to$

$\log (|\begin{matrix} 1 + \cot (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 - \tan (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 - \cot (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 + \tan (\frac{x}{2}) \end{matrix}|) + c$

624.

$\int \frac{dx}{7 + 5 \cos (x)} is equal to$

$\frac{1}{\sqrt{3}} \tan^{- 1} (\frac{1}{\sqrt{3}} \tan \frac{x}{2}) + c$
$\frac{1}{\sqrt{6}} \tan^{- 1} (\frac{1}{\sqrt{6}} \tan \frac{x}{2}) + c$
$\frac{1}{7} \tan^{- 1} (\tan \frac{x}{2}) + c$
$\frac{1}{4} \tan^{- 1} (\tan \frac{x}{2}) + c$

625.

$\int \frac{3^{x} dx}{\sqrt{9^{x} - 1}} is equal to$

$\frac{1}{\log (3)} \log (|\begin{matrix} 3^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (3)} \log (|\begin{matrix} 3^{x} - \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (9)} \log (|\begin{matrix} 3^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (3)} \log (|\begin{matrix} 9^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$

626.

$\int_{2}^{3} \frac{dx}{x^{2} - x is equal to}$

$\log (\frac{2}{3})$
$\log (\frac{4}{3})$
$\log (\frac{8}{3})$
$\log (\frac{1}{4})$

627.
$\int_{\frac{- π}{2}}^{\frac{π}{2}} \sin^{4} (x) \cos^{6} (x) dx is equal to$

$\frac{3 π}{128}$

$\frac{3 π}{256}$

$\frac{3 π}{572}$

$\frac{3 π}{64}$

$\frac{3 π}{256}$

$\int_{\frac{- π}{2}}^{\frac{π}{2}} \sin^{4} (x) \cos^{6} (x) dx = 2 \int_{0}^{\frac{π}{2}} \sin^{4} (x) \cos^{6} (x) dx = \frac{2 \frac{4 + 1}{2}! \frac{6 + 1}{2}!}{2 \frac{4 + 6 + 2}{2}!} = \frac{\frac{3}{2} \frac{1}{2} \sqrt{π} \frac{5}{2} \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{π}}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{3 π}{256}$