Cauchy formula for repeated integration
Integration has an interesting property where given some continuous function f(x), we can ‘compress’ n many antiderivatives of f into a single integral. This property was discovered by Cauchy in 1823.
Integration has an interesting property where given some continuous function f(x), we can ‘compress’ n many antiderivatives of f into a single integral. This property was discovered by Cauchy in 1823.
Let f be a continuous function on the real line. Then any iterative integral of f can be expressed
\int_a^x \cdots \int_a^{x_3} \int_a^{x_2} f(x_1) dx_1 \cdots dx_n = \frac{1}{(n-1)!}\int_a^x (x-t)^{n-1}f(t)dt
The proof is given by induction. Let n=1 be the base case. Then
\begin{align*} \int_a^x f(x) dx &= \frac{1}{1}\int_a^x (x-t)^0f(t)dt\\ &= \int_a^x f(t) dt \end{align*}
So the base case is clearly true. Assuming this is true for n, we must prove the n+1 case. Using the Leibniz integral rule, note
\frac{d}{dx} \left[ \frac{1}{n!} \int_a^x (x-t)^n f(t) dt \right] = \frac{1}{(n-1)!} \int_a^x (x-t)^{n-1} f(t) dt.