# How can I confirm $\underbrace{\int \ldots \int}_{n} |x| dx = \frac{x^n |x|}{n+1}+C$?

So I was burnt out and also determined to identify the uncertain indispensable of the outright value function, $|x|$. Making use of assimilation by components ($u=|x|, dv=dx$, $dx = \text{sgn}(x)=\frac{|x|}{x}$), it can be revealed that $\displaystyle\int |x| dx = \frac{x |x|}{2}+C$.

Currently I determined to take the indispensable once more, locating that $\displaystyle\int\left(\int |x| dx \right) dx=\frac{x^2 |x|}{3}+C$. Proceeding, I located the pattern in the title, that the $n$th uncertain indispensable of $|x|$ is $\displaystyle\frac{x^n |x|}{n+1}+C$. Exists a means to confirm this basic outcome?

Make the monitoring that $|x| = \theta(x) x - \theta(-x) x$ for all actual $x$, where $\theta$ is the Heaviside function (reviewing to $1$ if the argument declares and also $0$ or else). It is recognized that \begin{eqnarray} \int \theta(x) \ x \ dx = \theta(x) \frac{x^{2}}{2} + C \quad \text{and} \quad \int \theta(-x) \ x \ dx = \theta(-x) \frac{x^{2}}{2} + C^{\prime}, \end{eqnarray} where $C$ and also $C^{\prime}$ are constants of assimilation. The identification for $n = 1$ adheres to by reduction and also the depiction of $|x|$ over. With $n$ assimilations, we have \begin{eqnarray} \int \cdots \int |x| \ dx = \theta(x) \int \cdots \int x \ dx - \theta(-x) \int \cdots \int x \ dx = \frac{|x| x^{n}}{(n+1)!} + P_{n}, \end{eqnarray} where $P_{n}$ is a polynomial in $x$, as asserted.

Yes, there is a means: Use Mathematical induction. I do not add even more information due to the fact that I assume it is primary, isn't it?

The $C$ in the title of this inquiry need to be changed by an approximate polynomial $p(x)$ of level $\le n-1$. Nevertheless, we are broaching an $n$ - layer assimilation below.

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