Mathematics

Let  $f$  and  $g$  be continuous functions on  $[ 0 , a]$  such that  $f ( x ) = f ( a - x )$  and  $g ( x ) + g ( a - x ) = 4,$  then   $\int _ { 0 } ^ { a } f ( x ) g ( x ) d x$  is equal to :-

$2 \int _ { 0 } ^ { a } f ( x ) d x$

SOLUTION
${ I }=\int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ g }({ x }){ dx }$

${ I }=\int _{ { 0 } }^{ { { a } } }{ f } ({ a }-{ x }){ g }({ a }-{ x }){ d }{ x }$

${ I }=\int _{ { 0 } }^{ { { a } } }{ f } ({ x })(4-{ g }({ x })){ d }{ x }$

${ I }=4\int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ d }{ x }-{ I }$

$\Rightarrow { I }=2\int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ d }{ x }$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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