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Integrals

Multiple Choice Questions

141.

The value of $\int_{1}^{4} |x - 3| dx$ is equal to :

1
$\frac{5}{2}$
$\frac{1}{2}$
$\frac{3}{2}$

142.

The value of $\int_{0}^{\infty} \frac{dx}{(a^{2} + x^{2})}$ is equal to

$\frac{π}{2}$
$\frac{π}{2 a}$
$\frac{π}{a}$
$\frac{1}{2 a}$

143.

The value of the integral $\int e^{x} {(\frac{1 - x}{1 + x^{2}})}^{2} dx$ is

$e^{x} (\frac{1 - x}{1 + x^{2}}) + c$
$e^{x} (\frac{1 + x}{1 + x^{2}})$
$\frac{e^{x}}{1 + x^{2}} + c$
$e^{x} (1 - x) + c$

144.

If g(x) = $\int_{0}^{x} \cos^{4} (t) d t$ , then $g (x + π)$ is equsl to

g(x) + g( $π$ )
g(x) - g( $π$ )
g(x) . g( $π$ )
$\frac{g (x)}{g (π)}$

145.
$\int (\sin^{6} (x) + \cos^{6} (x) + 3 \sin^{2} (x) \cos^{2} (x)) dx$ is equal to

x + C

$\frac{3}{2} \sin (2 x) + C$

$- \frac{3}{2} \cos (2 x) + C$

$\frac{1}{3} \sin (3 x) - \cos (3 x) + C$

x + C

$Let I = \int (\sin^{6} (x) + \cos^{6} (x) + 3 \sin^{2} (x) \cos^{2} (x)) = \int \{{(\sin^{2} (x))}^{3} + {(\cos^{2} (x))}^{3} + 3 \sin^{2} (x) \cos^{2} (x)\} dx = \int [(\sin^{2} (x) + \cos^{2} (x)) (\sin^{4} (x) + \cos^{4} (x) - \sin^{2} (x) \cos^{2} (x)) + 3 \sin^{2} (x) \cos^{2} (x)] = \int [{(\sin^{2} (x) + \cos^{2} (x))}^{2} - 3 \sin^{2} (x) \cos^{2} (x) + 3 \sin^{2} (x) \cos^{2} (x)] = \int 1 dx = x + C$