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Question-213656




Question Number 213656 by efronzo1 last updated on 12/Nov/24
Answered by golsendro last updated on 13/Nov/24
  (i) g(4−x)= −g(x)           ∫_0 ^4  g(x)dx = ∫_0 ^4  g(4−x)dx           ∫_0 ^4 −g(4−x)dx = ∫_0 ^4 g(4−x)dx        it folow that ∫_0 ^4  g(4−x)dx= 0       and ∫_0 ^4  g(x)dx = 0   (ii) f(x^2 )= f((−x)^2 ) , f(x) even function     ∫_(−4) ^4  f(x^2 )dx = 2∫_0 ^4  f(x^2 )dx                             = 2 ∫_0 ^4  (4x^3 −g(x))dx                             = 2∫_0 ^4  4x^3  dx−0                            = 2[ x^4  ]_0 ^4  = 2×256= 512
(i)g(4x)=g(x)40g(x)dx=40g(4x)dx40g(4x)dx=40g(4x)dxitfolowthat40g(4x)dx=0and40g(x)dx=0(ii)f(x2)=f((x)2),f(x)evenfunction44f(x2)dx=240f(x2)dx=240(4x3g(x))dx=2404x3dx0=2[x4]04=2×256=512

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