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]}
, it is
\displaystyle f'(x){\color{blue}g(x)} =
f[2] \cdot \left(f[1] \right)
= f[5]
.
(In \displaystyle \int g'(x) \; dx = g(x)
is the integration constant
C=0.
)
Verify
\displaystyle \int f[5] \; dx =
f[7] + C
with another partial integration.
It is
\displaystyle f(x){\color{blue}g(x)} =
f[0] \cdot \left(
{\color{blue}f[1]} \right).
Thus, together
\displaystyle \int f[4] \; dx = f[6] +
C
.