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

Using termwise division gives \(\displaystyle f(x) = - x - 1 + \frac{2}{x} + \frac{3}{x^{2}}\). Finding the antiderivative of each term gives \(\displaystyle F(x) = - \frac{x^{2}}{2} - x + 2 \ln{\left(x \right)} - \frac{3}{x} + C\)

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\begin{question}Find the anti-derivative of $f(x) = \frac{- x^{3} - x^{2} + 2 x + 3}{x^{2}}$
    \soln{9cm}{Using termwise division gives $f(x) = - x - 1 + \frac{2}{x} + \frac{3}{x^{2}}$. Finding the antiderivative of each term gives $F(x) = - \frac{x^{2}}{2} - x + 2 \ln{\left(x \right)} - \frac{3}{x} + C$}

\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{- x^{3} - x^{2} + 2 x + 3}{x^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20%2B%202%20x%20%2B%203%7D%7Bx%5E%7B2%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = \frac{- x^{3} - x^{2} + 2 x + 3}{x^{2}} " data-equation-content=" \displaystyle f(x) = \frac{- x^{3} - x^{2} + 2 x + 3}{x^{2}} " /> </p> </p>
HTML for Canvas
<p> <p>Using termwise division gives  <img class="equation_image" title=" \displaystyle f(x) = - x - 1 + \frac{2}{x} + \frac{3}{x^{2}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%20x%20-%201%20%2B%20%5Cfrac%7B2%7D%7Bx%7D%20%2B%20%5Cfrac%7B3%7D%7Bx%5E%7B2%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = - x - 1 + \frac{2}{x} + \frac{3}{x^{2}} " data-equation-content=" \displaystyle f(x) = - x - 1 + \frac{2}{x} + \frac{3}{x^{2}} " /> . Finding the antiderivative of each term gives  <img class="equation_image" title=" \displaystyle F(x) = - \frac{x^{2}}{2} - x + 2 \ln{\left(x \right)} - \frac{3}{x} + C " src="/equation_images/%20%5Cdisplaystyle%20F%28x%29%20%3D%20-%20%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%7D%20-%20x%20%2B%202%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%20-%20%5Cfrac%7B3%7D%7Bx%7D%20%2B%20C%20" alt="LaTeX:  \displaystyle F(x) = - \frac{x^{2}}{2} - x + 2 \ln{\left(x \right)} - \frac{3}{x} + C " data-equation-content=" \displaystyle F(x) = - \frac{x^{2}}{2} - x + 2 \ln{\left(x \right)} - \frac{3}{x} + C " /> </p> </p>