022 Exam 2 Sample A, Problem 2

Find the antiderivative of  $y\,=\,3x^{2}-12x+8.$

Foundations:
We only require some fundamental rules for antiderivatives/integrals. We have the power rule:
$\int x^{n}\,dx\,=\,{\frac {x^{n+1}}{n+1}}+C,$  for $x\neq -1.$
Also, like derivatives, multiplication by a constant and addition/subtraction are respected by the antiderivative:
$\int c\cdot f(x)+g(x)\,dx\,=\,c\int f(x)\,dx+\int g(x)\,dx.$

Solution:
We can apply the rules listed above to find
${\begin{array}{rcl}\int y\,dx&=&\int 3x^{2}-12x+8\,dx\\\\&=&3\cdot {\frac {x^{3}}{3}}-12\cdot {\frac {x^{2}}{2}}+8x+C\\\\&=&x^{3}-6x^{2}+8x+C.\end{array}}$
Do not forget the constant when evaluating an antiderivative (i.e., an integral without upper and lower bounds)!

Final Answer:
$x^{3}-6x^{2}+8x+C.$

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