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Let f(x) be a continuous and differentia...

Let `f(x)` be a continuous and differentiable function such that `f(x)=int_0^xsin(t^2-t+x)dt` Then prove that `f^('')(x)+f(x)=cosx^2+2xsinx^2`

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`f(x)=int_(0)^(x)sin(t^(2)-t+x)dt`
`=cosx int_(0)^(x)sin(t^(2)-t)dt+sinx int_(0)^(x)cos(t^(2)-t)dt`.
or `f'(x)=-sin int_(0)^(x)sin(t^(2)-t)dt+cosx sin (x^(2)-x)`
`+cosx int_(0)^(x)cos(t^(2)-t)dt+sinx cos (x^(2)-x)`
`=-sinx int_(0)^(x)sin(t^(2)-t)dt+cosx int_(0)^(x)cos(t^(2)-t)dt+sinx^(2)`
or `f''(x)=-sinx sin(x^(2)-x)-cosx int_(0)^(x)sin(t^(2)-t)dt-sinx`
`int_(0)^(x)cos(t^(2)-t)dt+cosxcos(x^(2)-x)+2xsinx^(2)`
`=cosx^(2)-f(x)+2xsinx^(2)`
or `f''(x)+f(x)=cosx^(2)+2xsinx^(2)`
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