Mathematics

Problem

A student evaluates

\displaystyle \int x^{4}\sin x\,dx

by writing successive derivatives of

x^{4}

and successive antiderivatives of

\sin x

in parallel columns, then combining the products with alternating signs. Which statement best explains why this process ends after a finite number of pairings? a) The derivatives of

x^{4}

eventually cycle back to

x^{4}

, closing the process. b) The derivatives of

x^{4}

eventually become zero, so no further non-zero products are formed. c) The derivatives of

\sin x

eventually become zero, eliminating new terms. d) Because there can be at most four pairings—the degree of

x^{4}

—the tabular pattern itself stops even though the functions remain non-zero.

Intuitive explanation

This is about integration by parts and why the tabular method works. The key insight is that each derivative of $x^4$ reduces the power by one, so after exactly four differentiations you reach zero—and once you hit zero, all subsequent rows contribute nothing. The alternating signs and antiderivatives of $\sin x$ (which cycle through $\sin x, -\cos x, -\sin x, \cos x$ ) just tag along; the process terminates because polynomial derivatives eventually vanish, not because of anything special about the trig function.

Intuitive explanation

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