How to convert from multiple integrals to single integral I want to solve following integral $$\int_0^t\cdots \int_0^t (x_1+x_2\cdots x_M)dx_1dx_2\cdots dx_M$$ Is there some way through which I could convert these multiple integrals to single integral (for example through change of variable like $y=x_1+x_2\cdots x_M$). Any help in this regard is much appreciated.
 A: You can't reduce this down to a single integral, but you can switch variables so that the integrand depends on one variable only.  Here's how to do it:
We define new variables $y_i$ such that
\begin{align*}
y_1 &= x_1 + x_2 + \dots + x_M \\
y_2 &= x_2 \\
y_3 &= x_3 \\
&\vdots \\
y_M &= x_M
\end{align*}
The inverse transformations are
\begin{align*}
x_1 &= y_1 - y_2 - y_3 - \dots - y_M \\
x_2 &= y_2 \\
 &\vdots \\
x_M &= y_M
\end{align*}
and so the Jacobian matrix is
$$
\frac{\partial x_j}{\partial y_i} =
\begin{bmatrix} 1 & -1 & -1 &  & -1 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 &  & 0 \\
& \vdots & & \ddots & \vdots \\
0 & 0 & 0 & \dots & 1 \end{bmatrix}
$$
The determinant of this matrix is then 1, and so we have
$$
\idotsint f(x_1 + \dots + x_M) dx_1 dx_2 \cdots dx_M = \idotsint f(y_1) \left| \frac{\partial x_j}{\partial y_i} \right| dy_1 dy_2 \cdots dy_M \\= \idotsint f(y_1) dy_1 dy_2 \cdots dy_M.
$$
At this point, if the limits of integration for $y_1$ are independent of all the other $y_i$'s, one could factor this out as
$$
\left[ \int f(y_1) dy_1 \right] \left[\idotsint  dy_2 \cdots dy_M \right]
$$
and now you only have one non-trivial integral to perform (since the multiple integral over $y_2$ through $y_M$ is the integral of a constant.)  Whether or not you can do this will depend on the precise form of your region of integration.

EDIT: Somehow I missed that you did in fact provide concrete limits of integration:  for all $x_i$, we have $0 \leq x_i \leq t$.  To translate these into limits on the $y_i$, we first note that obviously we must have $0 \leq y_i \leq t$ for $i \geq 2$, since in these cases $y_i = x_i$.  But the integral over $y_1$ is a little more subtle.  Since $y_1 = x_1 + x_2 + \cdots + x_M = x_1 + y_2 + \cdots + y_M$, the range of allowable $y_1$ for a given $y_2, \dots, y_M$ is
$$
y_2 + \cdots + y_M \leq y_1 \leq t + y_2 + \cdots + y_M, 
$$
and so the integral in terms of the $y$ variables becomes
$$
\int_0^t \int_0^t \cdots \int_0^t \left[ \int_{y_2 + \cdots + y_M}^{t + y_2 + \cdots + y_m} f(y_1) dy_1 \right] dy_2 \cdots dy_M.
$$
Note that the limits of integration for $y_1$ are not independent of the other $y_i$.  This means that in general, the result of the $y_1$ integration will be a complicated function of $y_2$ through $y_M$, which you will then have to integrate with respect to all of the other variables.
You might be able to do a clever change of order of integration to make this more tractable, but I don't immediately see how.  For the specific case of $f(y_1) = y_1$, the technique proposed by @Med is probably better;  for a general $f(y_1)$, you might be stuck with doing $M$ integrals.
A: $\int_0^t\cdots \int_0^t (x_1+x_2\cdots x_M)dx_1dx_2\cdots dx_M=\sum_{i=1}^{M}\int_0^t\cdots \int_0^t (x_i)dx_1dx_2\cdots dx_M$
For each term in the summation, first take the integral with respect to $dx_i$ and then with respect to the rest of the variables.
$\sum_{i=1}^{M}\int_0^t\cdots \int_0^t (x_i)dx_1dx_2\cdots dx_M=M \times \frac{t^2}{2}\times t^{M-1}=\frac{Mt^{M+1}}{2}$
Let me know, if it is not what you are looking for.
A: You can use the concept applied in variation iteration method or variation of parameter method. This is done by multiplying the functional of the converted single integral with Lagrangian or Wronskian multiplier.
Let me know the outcome.
