022 Exam 2 Sample A, Problem 2

Find the antiderivative of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\,=\,3x^2-12x+8.}

Foundations:
We only require some fundamental rules for antiderivatives/integrals. We have the power rule:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^n\,dx\,=\,\frac{x^{n+1}}{n+1} +C,}   for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\neq -1.}
Also, like derivatives, multiplication by a constant and addition/subtraction are respected by the antiderivative:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \int y\, dx & = & \int3x^{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:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3-6x^2+8x+C.}

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