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Integrals

Multiple Choice Questions

11.

$What is \int_{0}^{1} x {(1 - x)}^{9} d x = ?$

$\frac{1}{110}$
$\frac{1}{132}$
$\frac{1}{148}$
$\frac{1}{240}$

12.

$What is \int_{2}^{8} |x - 5| d x = ?$

13.

What is $\int \sin^{3} (x) \cos (x) dx = ?$

$\cos^{4} (x) + c$
$\sin^{4} (x) + c$
$\frac{{(1 - \sin^{2} (x))}^{2}}{4} + c$
$\frac{{(1 - \cos^{2} (x))}^{2}}{4} + c$

14.

$What is \int e^{\ln (\tan (x))} = ?$

$\ln |\tan (x)| + c$
$\ln |\sec (x)|$
$\tan (x) + c$
$e^{\tan (x)} + c$

15.

$What is \int_{- 1}^{1} \frac{d}{dx} (\tan^{- 1} (\frac{1}{x})) d x = ?$

0
$- \frac{π}{4}$
$- \frac{π}{2}$
$\frac{π}{2}$

16.

$What is \int \frac{dx}{a^{2} \sin^{2} (x) + b^{2} \cos^{2} (x)} = ?$

$c + \frac{1}{ab} \tan^{- 1} (\frac{atan (x)}{b})$
$c - \frac{1}{ab} \tan^{- 1} (\frac{b \tan (x)}{a})$
$c + \frac{1}{ab} \tan^{- 1} (\frac{b \tan (x)}{a})$
None of the above

17.
$What is \int \frac{dx}{x (x^{7} = 1)} = ?$

$\frac{1}{2} \ln (|\frac{x^{7} - 1}{x^{7} + 1}|) + c$

$\frac{1}{7} \ln (\frac{x^{2} + 1}{x^{7}}) + c$

$\ln (|\frac{x^{2} - 1}{7 x}|) + c$

$\frac{1}{7} \ln (|\frac{x^{7}}{x^{7} + 1}|) + c$

$\frac{1}{7} \ln (|\frac{x^{7}}{x^{7} + 1}|) + c$

$\frac{1}{x (x^{7} + 1)} dx Putting x^{7} + 1 = t 7 x^{6} dx = dt \int \frac{1}{x (x^{7} + 1)} dx = \frac{1}{7} \int \frac{1}{t (t - 1)} dt = \frac{1}{7} \ln (\frac{t - 1}{t}) + c = \frac{1}{7} \ln (\frac{x^{7} 1 - 1}{x^{7} + 1}) + c = \frac{1}{7} \ln (\frac{x^{7}}{x^{7} + 1}) + c$