Consider the following integral:
$$\int 8 t^{3} + 6 t + 5\, dt$$
Recall the rules for taking the derivative of a polynomial: the derivative of $t^{7} + 4 t^{5} + 11 t$ is
. When we take the derivative of a polynomial, we treat each term separately. For each term, we multiply the coefficient by the
, and
. Does the order of these two operations matter?
If we want to undo taking the derivative of a polynomial, we again treat each term separately, and need to undo both operations in the
order. We need to
and then
.
Let's look at the first term: $8 t^{3}$. The new exponent is
, and the new coefficient is
.
For the second term, $6 t$, the new exponent is
, and the new coefficient is
.
For the third term, $5=5 t^{0}$, the new exponent is
, and the new coefficient is
.
Then $\displaystyle \int 8 t^{3} + 6 t + 5\, dt=$