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Byju's Answer

Standard XII

Mathematics

Property 4

fx gx d x=21 ...

Question

f(x)g(x) dx=21f(x) dx , iff and g are defined asf(x)=f(a-x)19. Show thatand g(x) + g(a-x) = 4

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Solution

The given integral is,

I= ∫ 0 a f( x )g( x )dx (1)

On solving integral, we get,

I= ∫ 0 a f( x )g( x )dx = ∫ 0 a f( a−x )g( a−x )dx (2)

Adding equation (1) and (2),

I+I= ∫ 0 a f( x )g( x )dx + ∫ 0 a f( x )g( a−x )dx 2I= ∫ 0 a f( x )( g( x )+g( a−x ) )dx

It is given that,

g(x)+g( a−x )=4.

2I= ∫ 0 a f( x )( g( x )+g( a−x ) )dx 2I= ∫ 0 a 4×f( x )dx I=2 ∫ 0 a f( x )dx

Hence, it is proved that ∫ 0 a f( x )g( x )dx =2 ∫ 0 a f( x )dx .

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