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 :-


ANSWER

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


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
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