022 Exam 2 Sample A, Problem 4

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 the derivative of the natural log since we will recover natural log from integration:
$\left(ln(x)\right)'={\frac {1}{x}}$
Finally, we will need our power rule for integration:
$\int x^{n}dx\,=\,{\frac {x^{n+1}}{n+1}},$ for $n\neq 0$,

 Solution:

Step 1:
Use a U-substitution with $u=3x+2.$ This means $du=3dx$, and after substitution we have $\int \left(3x+2\right)^{4}dx=\int u^{4}du$

Step 2:
We can no apply the power rule for integration: $\int u^{4}du={\frac {u^{5}}{5}}$

Step 3:
Since our original function is a function of x, we must substitute x back into the result from problem 2:
${\frac {u^{5}}{5}}={\frac {(3x+2)^{5}}{5}}$

Step 4:
As will all indefinite integrals, don't forget the "+C" at the end.

Final Answer:
$\int \left(3x+2\right)^{5}dx\,=\,{\frac {(3x+2)^{5}}{5}}+C$

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