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Calculus
Anti-Derivatives
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Find the anti-derivative of \(\displaystyle f(x) = \frac{- 6 x^{3} - 5 x^{2} + 8 x - 4}{x^{2}}\)

Using termwise division gives \(\displaystyle f(x) = - 6 x - 5 + \frac{8}{x} - \frac{4}{x^{2}}\). Finding the antiderivative of each term gives \(\displaystyle F(x) = - 3 x^{2} - 5 x + 8 \ln{\left(x \right)} + \frac{4}{x} + C\)

Download \(\LaTeX\) 

\begin{question}Find the anti-derivative of $f(x) = \frac{- 6 x^{3} - 5 x^{2} + 8 x - 4}{x^{2}}$
    \soln{9cm}{Using termwise division gives $f(x) = - 6 x - 5 + \frac{8}{x} - \frac{4}{x^{2}}$. Finding the antiderivative of each term gives $F(x) = - 3 x^{2} - 5 x + 8 \ln{\left(x \right)} + \frac{4}{x} + C$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the anti-derivative of  <img class="equation_image" title=" \displaystyle f(x) = \frac{- 6 x^{3} - 5 x^{2} + 8 x - 4}{x^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%206%20x%5E%7B3%7D%20-%205%20x%5E%7B2%7D%20%2B%208%20x%20-%204%7D%7Bx%5E%7B2%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = \frac{- 6 x^{3} - 5 x^{2} + 8 x - 4}{x^{2}} " data-equation-content=" \displaystyle f(x) = \frac{- 6 x^{3} - 5 x^{2} + 8 x - 4}{x^{2}} " /> </p> </p>
HTML for Canvas
<p> <p>Using termwise division gives  <img class="equation_image" title=" \displaystyle f(x) = - 6 x - 5 + \frac{8}{x} - \frac{4}{x^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%206%20x%20-%205%20%2B%20%5Cfrac%7B8%7D%7Bx%7D%20-%20%5Cfrac%7B4%7D%7Bx%5E%7B2%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = - 6 x - 5 + \frac{8}{x} - \frac{4}{x^{2}} " data-equation-content=" \displaystyle f(x) = - 6 x - 5 + \frac{8}{x} - \frac{4}{x^{2}} " /> . Finding the antiderivative of each term gives  <img class="equation_image" title=" \displaystyle F(x) = - 3 x^{2} - 5 x + 8 \ln{\left(x \right)} + \frac{4}{x} + C " src="/equation_images/%20%5Cdisplaystyle%20F%28x%29%20%3D%20-%203%20x%5E%7B2%7D%20-%205%20x%20%2B%208%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20%2B%20%5Cfrac%7B4%7D%7Bx%7D%20%2B%20C%20" alt="LaTeX:  \displaystyle F(x) = - 3 x^{2} - 5 x + 8 \ln{\left(x \right)} + \frac{4}{x} + C " data-equation-content=" \displaystyle F(x) = - 3 x^{2} - 5 x + 8 \ln{\left(x \right)} + \frac{4}{x} + C " /> </p> </p>