Product menace

Calculus Level 3

Let $\int^2f''(x)\ dx^2= f(x)$ , where $f''(x)$ is the second derivative of $f(x)$ , where $\int^2 f''(x)\ dx^2$ denotes the second indefinite integral of $f''(x)$ , with integration constant $C=0$ . Find

$\large \prod_{k=2}^n \int^{k-1} k! x\ dx^{k-1}$

\frac{x^n}{n}

x^n

x^{\frac{n^2+n-2}{2}}

x^\frac{n(n+1)}{2}

x^{n-1}

1 solution

Chew-Seong Cheong
Oct 11, 2018

$\begin{aligned} P & = \prod_{\color{#3D99F6}k=2}^n \int^{k-1} k! x \ dx^{k-1} \\ & = \prod_{\color{#D61F06}k=1}^{n-1} \int^k (k+1)! x \ dx^k \\ & = \int 2!x\ dx \int^2 3!x\ dx^2 \int^3 4!x\ dx^3 \cdots \int^{n-1} n!x\ dx^{n-1} \\ & = x^2 \cdot x^3 \cdot x^4 \cdots x^n \\ & = x^{\frac {(n-1)(n+2)}2} = \boxed{x^{\frac{n^2+n-2}2}} \end{aligned}$

Product menace

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