Determine \displaystyle \int f[4] \; dx.
Use C as the integration constant.
For partial integration, we use that
\displaystyle
\int f(x)g'(x)\; dx = f(x)g(x) - \int f'(x)g(x) \; dx + C
.
An appropriate choice of f and g should ensure
to find a primitive of
f'(x)g(x) in an easier way than as for f(x)g'(x).
Here, suitable choices are
\displaystyle f(x) = f[0] and
\displaystyle g'(x) = f[3].
With
\displaystyle f'(x) = f[2] and
\displaystyle
\color{blue}{g(x) = f[1]} is
\displaystyle f'(x){\color{blue}g(x)} =
f[2] \cdot \left({\color{blue}f[1]}\right)
= f[5] .
(In \displaystyle \int g'(x) \; dx = g(x)
is the integration constant
C=0.)
It is
\displaystyle f(x){\color{blue}g(x)} =
f[0] \cdot \left(
{\color{blue}f[1]} \right) =
f[1]f[0]
.
Determine \displaystyle \int f[5] \; dx and
and summarise
\displaystyle \int f[4] \; dx = f[6] +
C.