Let f x = ∫ 0 x g t d t , where g is a non-zero even function. If f x + 5 = g x , then ∫ 0 x f (…

Let fx=∫0xgtdt, where g is a non-zero even function. If fx+5=gx, then ∫0xf(t)dt equals
  1. ∫5x+5g(t)dt
  2.  ∫x+55g(t)dt
  3. 5∫x+55g(t)dt
  4. 2∫5x+5g(t)dt

Solution

Given: gx is an even function, then
fx=∫0xg(t)dt will be an odd function,
⇒f-x=-fx
Let I=∫0xftdt put t=z+5
⇒I=∫-5x-5fz+5dz
⇒I=∫-5x-5g(z)dz   ∵fx+5=gx
⇒I=∫-5x-5f'(z)dz                    (∵f'(x)=g(x))
⇒I=f(z)|-5x-5=f(x-5)-f-5
⇒I=f(5)-f5+x ( f is an odd function and g is given as an even function)
⇒I=∫5+x5f'(t)dt=∫5+x5g(t)dt

Asked in: JEE Main 2019 (08 Apr Shift 2)

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