# 022 Exam 2 Sample A, Problem 3

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Find the antiderivative of $\int {\frac {1}{3x+2}}\,dx.$ Foundations:
This problem requires two rules of integration. In particular, you need
Integration by substitution (U - sub): If $f$ and $g$ are differentiable functions, then

$(f\circ g)'(x)=f'(g(x))\cdot g'(x).$ The Product Rule: If $f$ and $g$ are differentiable functions, then

$(fg)'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x).$ The Quotient Rule: If $f$ and $g$ are differentiable functions and $g(x)\neq 0$ , then

$\left({\frac {f}{g}}\right)'(x)={\frac {f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^{2}}}.$ Additionally, we will need our power rule for differentiation:
$\left(x^{n}\right)'\,=\,nx^{n-1},$ for $n\neq 0$ ,
as well as the derivative of natural log:
$\left(\ln x\right)'\,=\,{\frac {1}{x}}.$ Solution:

Step 1:
Use a U-substitution with $u=3x+2.$ This means $du=3dx$ , and after substitution we have
$\int {\frac {1}{3x+2}}=\int {\frac {1}{3u}}du$ Step 2:
We can now take the integral remembering the special rule:
$\int {\frac {1}{3u}}du={\frac {\log(u)}{3}}$ Step 3:
Since this integral is an indefinite integral we have to remember to add C at the end.
$\int {\frac {1}{3x+2}}dx={\frac {\ln(3x+2)}{3}}$ 