Please help to solve this problem
 

Asked by rajuharan | 26th Jan, 2017, 07:47: PM

Expert Answer:

begin mathsize 16px style Let space straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space space space...... left parenthesis straight i right parenthesis
We space know space that comma space integral subscript straight a superscript straight b space space straight f left parenthesis straight x right parenthesis space dx equals space integral subscript straight a superscript straight b space space straight f left parenthesis straight a plus straight b minus straight x right parenthesis space dx
So comma space straight I space equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator left parenthesis 2 plus left parenthesis negative 2 right parenthesis minus straight x right parenthesis squared over denominator 1 plus 5 to the power of 2 plus left parenthesis negative 2 right parenthesis minus straight x end exponent end fraction dx
rightwards double arrow straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator left parenthesis 2 plus left parenthesis negative 2 right parenthesis minus straight x right parenthesis squared over denominator 1 plus 5 to the power of 2 plus left parenthesis negative 2 right parenthesis minus straight x end exponent end fraction dx
rightwards double arrow straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator left parenthesis negative straight x right parenthesis squared over denominator 1 plus 5 to the power of negative straight x end exponent end fraction dx
rightwards double arrow straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared over denominator 1 plus 5 to the power of negative straight x end exponent end fraction dx space space space...... left parenthesis ii right parenthesis
Adding space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
2 straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx plus integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared over denominator 1 plus 5 to the power of negative straight x end exponent end fraction dx
rightwards double arrow 2 straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx plus integral subscript negative 2 end subscript superscript 2 space space fraction numerator 5 to the power of straight x straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx
rightwards double arrow 2 straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared plus 5 to the power of straight x straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space
rightwards double arrow 2 straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared left parenthesis 1 plus 5 to the power of straight x right parenthesis over denominator 1 plus 5 to the power of straight x end fraction dx space
rightwards double arrow 2 straight I equals integral subscript negative 2 end subscript superscript 2 space space fraction numerator straight x squared left parenthesis 1 plus 5 to the power of straight x right parenthesis over denominator 1 plus 5 to the power of straight x end fraction dx space
rightwards double arrow 2 straight I equals integral subscript negative 2 end subscript superscript 2 space space straight x squared dx
rightwards double arrow 2 straight I equals 2 integral subscript 0 superscript 2 space space straight x squared dx space.... left parenthesis Since space straight x squared space is space even right parenthesis
rightwards double arrow straight I equals integral subscript 0 superscript 2 space straight x squared dx
rightwards double arrow straight I equals right enclose straight x cubed over 3 end enclose squared subscript 0
rightwards double arrow straight I equals 2 cubed over 3 equals 8 over 3 end style

Answered by Rebecca Fernandes | 27th Jan, 2017, 04:36: PM