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Let f ( x) = int 0 ^ x g (t) d...

Let ` f ( x) = int _ 0 ^ x g (t) dt ` , where g is a non - zero even function . If ` f(x + 5 ) = g (x) ` , then ` int _ 0 ^ x f (t ) dt ` equals :

A

` 5 int _ ( x + 5 ) ^ 5 g (t) dt `

B

` 2 int _ 5 ^ ( x + 5) g (t) dt `

C

` int _ 5^ ( x + 5 ) g (t ) dt `

D

` int _ ( x + 5 )^( 5) g (t ) dt `

Text Solution

Verified by Experts

` f (x) = int _ 0 ^ x g (t ) dt `
` g (x) ` is even ` fn therefore f (x ) ` is odd `fn`.
Also ` f' (x) = g (x) , f (x + 5 ) = g (x) , f (5 - x ) = g ( - x ) = g (x ) = f ( x + 5 ) `
Now ` int _ 0 ^ x f (t ) dt `
` t = 5 + z " " dt = dz `
` int _ ( - 5 ) ^ ( x - 5 ) f (5 + z ) dz = int _ ( - 5 ) ^( x - 5 ) g (z ) dz = int _ ( - 5 ) ^( x - 5 ) f' (z) dz = f( x - 5 ) - f ( - 5 ) `
` = f (5) - f (5 - x) = f ( 5 ) - f (5 + x ) `
` int _ (x + 5) ^( 5) f' ( t ) dt = int _ (x + 5) ^( 5 ) g (t ) dt `
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