How to convert from multiple integrals to single integral

Question

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.

Answer 1

$\int_0^t\cdots \int_0^t (x_1+x_2\cdots x_M)dx_1dx_2\cdots dx_M=\sum_{i=1}^{M}\int_0^t\cdots \int_0^t (x_i)dx_1dx_2\cdots dx_M$

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.

Answer 2

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.

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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.