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Questions: Algebra BusinessCalculus
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Find the anti-derivative of \(\displaystyle f(x) = \frac{5 x^{3} + 9 x^{2} + 6 x + 6}{x^{2}}\)
Using termwise division gives \(\displaystyle f(x) = 5 x + 9 + \frac{6}{x} + \frac{6}{x^{2}}\). Finding the antiderivative of each term gives \(\displaystyle F(x) = \frac{5 x^{2}}{2} + 9 x + 6 \ln{\left(x \right)} - \frac{6}{x} + C\)
\begin{question}Find the anti-derivative of $f(x) = \frac{5 x^{3} + 9 x^{2} + 6 x + 6}{x^{2}}$
\soln{9cm}{Using termwise division gives $f(x) = 5 x + 9 + \frac{6}{x} + \frac{6}{x^{2}}$. Finding the antiderivative of each term gives $F(x) = \frac{5 x^{2}}{2} + 9 x + 6 \ln{\left(x \right)} - \frac{6}{x} + C$}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the anti-derivative of <img class="equation_image" title=" \displaystyle f(x) = \frac{5 x^{3} + 9 x^{2} + 6 x + 6}{x^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B5%20x%5E%7B3%7D%20%2B%209%20x%5E%7B2%7D%20%2B%206%20x%20%2B%206%7D%7Bx%5E%7B2%7D%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{5 x^{3} + 9 x^{2} + 6 x + 6}{x^{2}} " data-equation-content=" \displaystyle f(x) = \frac{5 x^{3} + 9 x^{2} + 6 x + 6}{x^{2}} " /> </p> </p><p> <p>Using termwise division gives <img class="equation_image" title=" \displaystyle f(x) = 5 x + 9 + \frac{6}{x} + \frac{6}{x^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%205%20x%20%2B%209%20%2B%20%5Cfrac%7B6%7D%7Bx%7D%20%2B%20%5Cfrac%7B6%7D%7Bx%5E%7B2%7D%7D%20" alt="LaTeX: \displaystyle f(x) = 5 x + 9 + \frac{6}{x} + \frac{6}{x^{2}} " data-equation-content=" \displaystyle f(x) = 5 x + 9 + \frac{6}{x} + \frac{6}{x^{2}} " /> . Finding the antiderivative of each term gives <img class="equation_image" title=" \displaystyle F(x) = \frac{5 x^{2}}{2} + 9 x + 6 \ln{\left(x \right)} - \frac{6}{x} + C " src="/equation_images/%20%5Cdisplaystyle%20F%28x%29%20%3D%20%5Cfrac%7B5%20x%5E%7B2%7D%7D%7B2%7D%20%2B%209%20x%20%2B%206%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cfrac%7B6%7D%7Bx%7D%20%2B%20C%20" alt="LaTeX: \displaystyle F(x) = \frac{5 x^{2}}{2} + 9 x + 6 \ln{\left(x \right)} - \frac{6}{x} + C " data-equation-content=" \displaystyle F(x) = \frac{5 x^{2}}{2} + 9 x + 6 \ln{\left(x \right)} - \frac{6}{x} + C " /> </p> </p>