Indefinite poly

Calculus Level 3

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

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

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

B C A

1 solution

Chew-Seong Cheong
Jul 16, 2017

$\begin{aligned} I & = \int P(x) g_0(x) \ dx & \small \color{#3D99F6} \text{By integration by parts} \\ & = P(x){\color{#3D99F6}g_1(x)} - \int P'(x) g_1(x) \ dx & \small \color{#3D99F6} \text{Note that }\int g_0(x) \ dx = g_1(x) \\ & = P(x)g_1(x) - P'(x) g_2(x) + \int P''(x) g_2(x) \ dx \\ & = P(x)g_1(x) - P'(x) g_2(x) + P''(x) g_3(x) - \cdots + (-1)^n {\color{#3D99F6}n!} g_{n+1}(x) & \small \color{#3D99F6} \text{Note that }P^{(n)}(x) = n! \end{aligned}$

Therefore, the answer is B .

Indefinite poly

1 solution

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