L’Hopital’s rule. Taylor’s Theorem and related results. Maclaurin series.
Basic integration: anti-derivatives, definite and indefinite integrals, methods of substitution and integration by parts.

这是一份drps.ed爱丁堡个大学MATH08073作业代写的成功案

The two results for $I$ and $J$ can be extended to any number of variables. Consider an $N$-dimensional Cartesian space, $\left{x_{i}: i=1,2, \ldots, N\right}$. Let $V_{N}$ be the volume in this space for which
$$
\sum_{i=1}^{N} x_{i} \leq 1 ; \quad x_{i}>0 \text { for } i=1,2, \ldots, N
$$
then
$$
I\left(m_{1}, m_{2}, \ldots, m_{N}\right) \equiv \int \cdots \int x_{1}^{m_{1}} x_{2}^{m_{2}} \ldots x_{N}^{m_{N}} d V_{N}
$$
where
$$
d V_{N}=\prod_{i=1}^{N} d x_{i} ; \quad m_{i}>-1 \text { for } i=1,2, \ldots, N
$$

MATH08073  COURSE NOTES ：

is given by the expression (see Franklin in $[4]$ ),
$$
I\left(m_{1}, m_{2}, \ldots, m_{N}\right)=\frac{\prod_{i=1}^{N} \Gamma\left(m_{i}+1\right)}{\Gamma\left(\sum_{i=1}^{N} m_{i}+N+1\right)} .
$$
Similarly, let $\bar{V}{N}$ be the volume for which $$ \sum{i=1}^{N}\left(\frac{x_{i}}{a_{i}}\right)^{p_{i}} \leq 1 ; \quad x_{i}>0 \text { for } i=1,2, \ldots, N
$$
where
$$
d V_{N}=\prod_{i=1}^{N} d x_{i} ; \quad m_{i}>-1 \text { for } i=1,2, \ldots, N
$$