First slide

Introduction to definite integration

Question

The value of $α$ for which $4 α \int_{- 1}^{2} e^{- α |x|} d x = 5$ , is:

Moderate

A

$\log_{e} (\frac{4}{3})$
B

$\log_{e} (\frac{3}{2})$
C

$\log_{e} \sqrt{2}$
D

$\log_{e} 2$

Solution

$4 α \{\int_{- 1}^{0} e^{α x} d x + \int_{0}^{2} e^{- α x} d x\} = 5$
$\Rightarrow 4 α \{{(\frac{e^{α x}}{α})}_{- 1}^{0} + \frac{e^{- α x}}{- α}|\} = 5$
$\Rightarrow 4 α \{(\frac{1 - e^{- α}}{α}) - (\frac{e^{- 2 α} - 1}{α})\} = 5$
$\Rightarrow 4 (2 - e^{- α} - e^{- 2 α}) = 5$
put $e^{- α} = t$
$\Rightarrow 4 t^{2} + 4 t - 3 = 0$ $\Rightarrow (2 t + 3) (2 t - 1) = 0$ $\Rightarrow e^{- α} = \frac{1}{2}$ $\Rightarrow α = \ln 2$

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