1.

Let f(x)=int_(0)^(x)f(t) dt equals

Answer»

`5int_(X+5)^(5)g(t)dt`
`int_(5)^(x+5)F(t)dt`
`2int_(5)^(x+5)f(t)dt`
`int_(x+5)^(5)g(t)dt`

Solution :Given , f (x) `=int_(0)^(x)f(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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