Question

integral -1 to1 e^|x|
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Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm :\longmapsto\:\displaystyle\int_{ - 1}^1\rm {e}^{ |x| } \: dx$

Let us assume that

$\rm :\longmapsto\:I \: = \: \displaystyle\int_{ - 1}^1\rm {e}^{ |x| } \: dx$

We know,

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\: |x| = \begin{cases} &\sf{ - x \: \: if \: x \: < \: 0} \\ &\sf{x \: \: if \: x \: \geqslant \: 0} \end{cases}\end{gathered}\end{gathered}$

So, given integral can be rewritten as

$\rm :\longmapsto\:I \: = \: \displaystyle\int_{ - 1}^0\rm {e}^{ |x| } \: dx + \:\displaystyle\int_0^1\rm {e}^{ |x| } \: dx$

$\rm :\longmapsto\:I \: = \: \displaystyle\int_{ - 1}^0\rm {e}^{ - x } \: dx + \:\displaystyle\int_0^1\rm {e}^{x} \: dx$

We know,

$\boxed{ \bf{\displaystyle\int\rm {e}^{x} dx \: = \: {e}^{x} + c }}$

So, above integral can be rewritten as

$\rm :\longmapsto\:I =\bigg[\dfrac{ {e}^{ - x} }{ - 1} \bigg]_0^1 + \bigg[ {e}^{x} \bigg]_0^1$

$\rm :\longmapsto\:I = - ( {e}^{0} - {e}^{ - ( - 1)}) + ( {e}^{1} - {e}^{0})$

$\rm :\longmapsto\:I = - ( {e}^{0} - {e}^{1}) + ( {e}^{1} - {e}^{0})$

$\rm :\longmapsto\:I = ( {e}^{1} - {e}^{0}) + ( {e}^{1} - {e}^{0})$

$\rm :\longmapsto\:I = 2( {e} - 1)$

Additional Information :-

$\boxed{ \bf{ \displaystyle\int_a^b\rm \: f(x) \: dx =\displaystyle\int_a^b\rm \: f(y) \: dy }}$

$\boxed{ \bf{ \displaystyle\int_a^b\rm \: f(x) \: dx = - \displaystyle\int_b^a\rm \: f(x) \: dx }}$

$\boxed{ \bf{ \displaystyle\int_0^a\rm \: f(x) \: dx = \displaystyle\int_0^a\rm \: f(a - x) \: dx }}$

$\boxed{ \bf{ \displaystyle\int_a^b\rm \: f(x) \: dx =\displaystyle\int_a^b\rm \: f(a + b - x) \: dx }}$

$\boxed{ \bf{ \displaystyle\int_0^{2a}\rm \: f(x) \: dx = 2\displaystyle\int_0^a\rm \: f(x) \: dx \: if \: f(2a - x) = f(x) }}$

$\boxed{ \bf{ \displaystyle\int_0^{2a}\rm \: f(x) \: dx = 0 \: if \: f(2a - x) = - f(x) }}$

$\boxed{ \bf{ \displaystyle\int_{ - a}^{a}\rm \: f(x) \: dx = 0 \: if \: f( - x) = - f(x) }}$

$\boxed{ \bf{ \displaystyle\int_{ - a}^{a}\rm \: f(x) \: dx = 2 \displaystyle\int_0^a\rm \: f(x) \: dx \: if \: f( - x) = f(x) }}$

Answer 2

Step-by-step explanation:

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