022 Exam 2 Sample A, Problem 4

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Find the antiderivative of $\int (3x+2)^{4}\,dx.$ Foundations:
This problem requires three rules of integration. In particular, you need
Integration by substitution (u - sub): If $u=g(x)$ is a differentiable functions whose range is in the domain of $f$ , then
$\int g'(x)f(g(x))dx=\int f(u)du.$ We also need our power rule for integration:
$\int x^{n}dx\,=\,{\frac {x^{n+1}}{n+1}}+C,$ for $n\neq 0$ .

Solution:

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