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If $f(x) = \cos x –\int_0^x(x-t)f(t)dt$, then $f’’ (x) + f(x)$ equals |
$–\cos x$ 0 $\int_0^x(x-t)f(t)dt$ $-\int_0^{-x}(x-t)f(t)dt$ |
$–\cos x$ |
$f(x) = \cos x –\int_0^x(x-t)\,f(t)\,dt$ $f(x) = \cos x –x\int_0^xf(t)\,dt+\int_0^xt\,f(t)dt$ $∴ f’(x) = –\sin x –\left\{xf(x)+\int_0^xf(t)\,dt\right\}+ x f (x)$ $= –\sin x –\int_0^xf(t)\,dt$ $∴f’’(x) = –\cos x – f(x) ⇒ f’’(x) + f(x) = –\cos x$ |
