Let \(f\) be a differentiable function such that \(\displaystyle \int f(x)\sin{x}  dx = -f(x)\cos{x} + \int 4x^3\cos{x}   dx \). Which of the following could be \(f(x)\) ?

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The equation shown reflects integration by parts:

$$ \int u   dv = uv - \int v   du $$ $$ \int f(x)\sin{x}  dx = -f(x)\cos{x} - \int -4x^3\cos{x}   dx $$ $$ u=f(x) \hskip{2em} dv=\sin{x}  dx$$ $$ du=f'(x) \hskip{2em} v=-\cos{x} $$

Equation the last portion (\(\int v   du \)):

$$f'(x)\cdot -\cos{x} = -4x^3\cos{x}$$ $$ f'(x)=4x^3 $$ $$ f(x)=\boxed{x^4} $$