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Let f(x)=int_(0)^(x)f(t) dt equals |
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Answer» `5int_(X+5)^(5)g(t)dt` On replacingx by (-x) , we GET `f(x)=int_(0)^(-x)g (t)dt` Now , PUT t =- u , so `int(-x)=-int_(0)^(x)g(-u)du=-int_(0)^(x)g(u)=-f(x)` `[:' `g is an even function] `rArrf(-x)=-f(xrArr f` an odd function. Now , it is given that f (x +5) = g (x) `:.f(5-x)=g(-x)=g(x)=f(x+5)` [ `:.` g is an even function] `rArrf(5-x)=f(x+5)`. . .(i) Let `I=int_(0)^(x)f(t)dt` Putt = u +5 `rArrt -5=u rArrdt = du` `:.u=-trArrdu=-dt ` , we get `I=-int_(5)^(5-x)g(-t)dt=int_(5-x)^(5)f (t) dt` `[:'-int_(a)^(b)f(x)dx=int_(b)^(a)f (x)` dx andg is an even function] `I=int_(5-x)^(5)f'(t)dt` [ by Leibnitz RULE f ' (x) = g (x) ] `=f(5)-f(5-x)=f(5)-f(5+x)`[ from Eq . (i)] `=int_(5+x)^(5) f ' (t) dt = int_(5+x)^(5)g (t) dt` |
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