# 022 Exam 2 Sample A, Problem 4

Find the antiderivative of ${\displaystyle \int (3x+2)^{4}\,dx.}$

Foundations:
This problem requires three rules of integration. In particular, you need
Integration by substitution (u - sub): If ${\displaystyle u=g(x)}$  is a differentiable functions whose range is in the domain of ${\displaystyle f}$, then
${\displaystyle \int g'(x)f(g(x))dx=\int f(u)du.}$
We also need our power rule for integration:
${\displaystyle \int x^{n}dx\,=\,{\frac {x^{n+1}}{n+1}}+C,}$  for ${\displaystyle n\neq 0}$.

Solution:

Step 1:
Use a u-substitution with ${\displaystyle u=3x+2.}$ This means ${\displaystyle du=3\,dx}$, or ${\displaystyle dx=du/3}$. After substitution we have
${\displaystyle \int \left(3x+2\right)^{4}\,dx\,=\,\int u^{4}\,{\frac {du}{3}}\,=\,{\frac {1}{3}}\int u^{4}\,du.}$
Step 2:
We can no apply the power rule for integration:
${\displaystyle {\frac {1}{3}}\int u^{4}\,du\,=\,{\frac {1}{3}}\cdot {\frac {u^{5}}{5}}\,=\,{\frac {u^{5}}{15}}.}$
Step 3:
Since our original function is a function of ${\displaystyle x}$, we must substitute ${\displaystyle x}$ back into the result from step 2:
${\displaystyle {\frac {u^{5}}{5}}\,=\,{\frac {(3x+2)^{5}}{5}}.}$
Step 4:
As will all indefinite integrals, don't forget the constant  ${\displaystyle C}$ at the end.
${\displaystyle \int \left(3x+2\right)^{4}\,dx\,=\,{\frac {(3x+2)^{5}}{15}}+C.}$