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 ($\int v   du$):

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