# Math Help - Integration by limit of sum

1. ## Integration by limit of sum

Hello...

I need to calculate this integral using the limit of the summation.

The period [0,T] is split into N intervals and the value point is at the left ot the interval.

$
\int_0^{T} x^3 (t)\ dX(t)\ = \lim_{N\to \infty}\sum_{i=0}^{N-1} X_{i}^{3}(X_{i+1} - X_{i})
$

Many Thanks to Galactus for his yesterday's idea but I believe my problem was not well posed.

I know the solution for something very similar:

$
\int_0^{T} 2X(t)\ dX(t)\ = \lim_{N\to \infty}\sum_{i=0}^{N-1} 2X_i (X_{i+1} - X_{i}) = X^2(T) - T
$

This is verifiable using the trick 2ab= (a+b)^2 -a^2 -b^2 and making a=X(i), b=X(i+1)-X(i)
Straight, basic telescopic series come out and give the result.