Integration

To integrate: Add one to the power and divide by the new power. When integrating always add a constant,
$c$
.
$4x^3$
integrated is
$\int 4 x^{3}dx=\frac{4x^{3+1}}{3+1}+c = x^4+c$
.
The
$\int$
symbol means integrate and the
$dx$
above means integrate with respect to
$x$
.
We can integrate a sum using the same rule for each term.
$2x^5-4x^7$
when integrated is
$\int 2x^5-4x^7 dx = \frac{2x^{5+1}}{5+1}\frac{4x^{7+1}}{7+1}+c = \frac{x^6}{3}-\frac{x^8}{2}+c$
.
This rule 'add one to the power and divide by the new power' works for
$x$
's and constants too.
To integrate
$3x$
write as
$3x^1$
then apply the rule to give
$\frac 3x^1 dx = \frac{3x^{1+1}}{1+1}+c = \frac{3x^2}{2}+c$
.
To integrate
$4$
write as
$4x^0$
then integrate using the above gives
$\int 4x^0 dx = \frac{4x^{0+1}}{0+1}+c=4x+c$
.
Integrate
$4x^2-6x-4$
.
Write
$4x^2-6x^1-4x^0$
.
We have
\begin{aligned} \int 4x^2-6x^1-4x^0 dx &= \frac{4x^{2+1}}{2+1}- \frac{6x^{1+1}}{1+1}- \frac{4x^{0+1}}{0+1}+c \\ &= \frac{4x^3}{3}-3x^2-4x+c \end{aligned}