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If f(x)=cosx-int0^x(x-t)f(t)dt ,t h e nf...

If `f(x)=cosx-int_0^x(x-t)f(t)dt ,t h e nf^(prime)(x)+f(x)` is equal to `-cosx` (b) `-sinx` `int_0^x(x-t)f(t)dt` (d) 0

A

`-cosx`

B

`-sinx`

C

`int_(0)^(x)(x-t)f(t)dt`

D

`0`

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x)=cosx=int_(0)^(x)(x-t)f(t)dt`
`=cosx-x int_(0)^(x)f(t)dt+int_(0)^(x) tf(t)dt`
`:.f'(x)=-sinx-xf(x)-int_(0)^(x)f(t)dt+xf(x)`
`=-sinx-int_(0)^(x)f(t)dt`
`:. f''(x)=-cosx-f(x)`
or `f''(x)+f(x)=-cosx`
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