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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( $π$ )

$\frac{g (x)}{g (π)}$

g(x) + g( $π$ )

$∵ g (x) = \int_{0}^{x} \cos^{4} (t) d t ∴ g (x + π) = \int_{0}^{π + x} \cos^{4} (t) d t = \int_{0}^{π} \cos^{4} (t) d t + \int_{π}^{π + x} \cos^{4} (t) d t = I_{1} + I_{2}$

$\Rightarrow I_{1} = g (π) and I_{2} = \int_{π}^{π + x} \cos^{4} (t) d t \Rightarrow I_{2} = \int_{0}^{x} \cos^{4} (y + π) d y = \int_{0}^{x} {[\cos (π + y)]}^{4} dy = \int_{0}^{x} \cos^{4} (y) dy = g (x) ∴ g (π + x) = g (x) + g (π)$