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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. 1    
  3. 2    
  4. 3    

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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