\[c\]
.\[4x^3\]
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
\[\begin{equation} \begin{aligned} \int 4x^2-6x^1-4x^0 dx &= \frac{4x^{2+1}}{2+1}- \frac{6x^{1+1}}{1+1}- \frac{4x^{0+1}}{0+1}+c \\ &= \frac{4x^3}{3}-3x^2-4x+c \end{aligned} \end{equation}\]