Indefinite poly

Calculus Level 2

Let \(P(x)\) be a monic polynomial of degree \(n\) and \(g_0: \mathbb{R} \to \mathbb{R}\) be any integrable function such that

\[\begin{align} g_{1}(x) & = \int g_0(x) \ dx \\ g_{2}(x) & = \int g_1(x) \ dx \\ g_{3}(x) & = \int g_2(x) \ dx \\ & \quad \vdots \\ g_{n+1}(x) & = \int g_n(x) \ dx \end{align} \]

Then \(\displaystyle \int P(x) g_0(x) \ dx\) is equal to:

A: \(\quad n! + P'(x) + \displaystyle \int g_0(x) \ dx \)

B: \(\quad P(x) g_1(x) - P'(x) g_2(x) + \cdots + (-1)^nn!g_{n+1} (x) \)

C: \(\quad P(x) g_{1}(x) + P' (x) g_{2} (x) + P'' (x) g_{3}(x) + \cdots + n g_{n+1} (x) \)

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