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Maths-

General

Easy

Question

If f (x) = $open curly brackets table row cell e to the power of cos invisible function application x end exponent end cell cell sin invisible function application x text for end text vertical line x vertical line less or equal than 2 end cell row cell text 2, end text end cell cell text otherwise end text end cell end table comma text then end text not stretchy integral subscript text -2 end text end subscript superscript 3 end superscript f left parenthesis x right parenthesis d x equals close$

-1
1
2
3

hint Hint:

$integral subscript a superscript b f x d x space equals space integral subscript a superscript c f x d x space plus integral subscript c superscript b f x d x$

The correct answer is: 2

Given, f (x) = $open curly brackets table row cell e to the power of cos invisible function application x end exponent end cell cell sin invisible function application x text for end text vertical line x vertical line less or equal than 2 end cell row cell text 2, end text end cell cell text otherwise end text end cell end table close$
To find : f (x) = $integral subscript text -2 end text end subscript superscript 3 f x d x$
By formula :
$integral subscript a superscript b f x d x space equals space integral subscript a superscript c f x d x space plus integral subscript c superscript b f x d x$
$integral subscript text -2 end text end subscript superscript 3 f x d x space equals space integral subscript negative 2 end subscript superscript 2 f x d x space plus integral subscript 2 superscript 3 f x d x integral subscript text -2 end text end subscript superscript 3 f x d x space equals space integral subscript negative 2 end subscript superscript 2 e to the power of cos x end exponent sin x d x space plus integral subscript 2 superscript 3 2 d x W h e n space w e space r e p l a c e space x space b y space minus x integral subscript text -2 end text end subscript superscript 3 f x d x space equals space integral subscript negative 2 end subscript superscript 2 e to the power of cos left parenthesis negative x right parenthesis end exponent sin left parenthesis negative x right parenthesis d x space plus integral subscript 2 superscript 3 2 d x integral subscript text -2 end text end subscript superscript 3 f left parenthesis negative x right parenthesis d x space equals space e to the power of cos x end exponent space. space left parenthesis negative sin x right parenthesis space plus integral subscript 2 superscript 3 2 d x W e space k n o w space t h a t space integral subscript negative a end subscript superscript a f x d x space equals space 0 comma space w h e n space f left parenthesis negative x right parenthesis space equals space minus f left parenthesis x right parenthesis i. e. integral subscript text -2 end text end subscript superscript 3 f left parenthesis negative x right parenthesis d x space equals space 0 space plus 2 left parenthesis 3 minus 2 right parenthesis integral subscript text -2 end text end subscript superscript 3 f left parenthesis negative x right parenthesis d x space equals space 2$

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