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Calculus
Anti-Derivatives
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Find the anti-derivative of \(\displaystyle f(x) = \frac{- 7 x^{3} + 4 x^{2} + 7 x + 2}{\sqrt[3]{x}}\)

Using termwise division gives \(\displaystyle f(x) = - 7 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} + 7 x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}}\). Finding the antiderivative of each term gives \(\displaystyle F(x) = - \frac{21 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} + \frac{21 x^{\frac{5}{3}}}{5} + 3 x^{\frac{2}{3}} + C\)

Download \(\LaTeX\) 

\begin{question}Find the anti-derivative of $f(x) = \frac{- 7 x^{3} + 4 x^{2} + 7 x + 2}{\sqrt[3]{x}}$
    \soln{9cm}{Using termwise division gives $f(x) = - 7 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} + 7 x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}}$. Finding the antiderivative of each term gives $F(x) = - \frac{21 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} + \frac{21 x^{\frac{5}{3}}}{5} + 3 x^{\frac{2}{3}} + C$}

\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{- 7 x^{3} + 4 x^{2} + 7 x + 2}{\sqrt[3]{x}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%207%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%207%20x%20%2B%202%7D%7B%5Csqrt%5B3%5D%7Bx%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = \frac{- 7 x^{3} + 4 x^{2} + 7 x + 2}{\sqrt[3]{x}} " data-equation-content=" \displaystyle f(x) = \frac{- 7 x^{3} + 4 x^{2} + 7 x + 2}{\sqrt[3]{x}} " /> </p> </p>
HTML for Canvas
<p> <p>Using termwise division gives  <img class="equation_image" title=" \displaystyle f(x) = - 7 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} + 7 x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%207%20x%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%20%2B%204%20x%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D%20%2B%207%20x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%2B%20%5Cfrac%7B2%7D%7B%5Csqrt%5B3%5D%7Bx%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = - 7 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} + 7 x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}} " data-equation-content=" \displaystyle f(x) = - 7 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} + 7 x^{\frac{2}{3}} + \frac{2}{\sqrt[3]{x}} " /> . Finding the antiderivative of each term gives  <img class="equation_image" title=" \displaystyle F(x) = - \frac{21 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} + \frac{21 x^{\frac{5}{3}}}{5} + 3 x^{\frac{2}{3}} + C " src="/equation_images/%20%5Cdisplaystyle%20F%28x%29%20%3D%20-%20%5Cfrac%7B21%20x%5E%7B%5Cfrac%7B11%7D%7B3%7D%7D%7D%7B11%7D%20%2B%20%5Cfrac%7B3%20x%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%7D%7B2%7D%20%2B%20%5Cfrac%7B21%20x%5E%7B%5Cfrac%7B5%7D%7B3%7D%7D%7D%7B5%7D%20%2B%203%20x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%20%2B%20C%20" alt="LaTeX:  \displaystyle F(x) = - \frac{21 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} + \frac{21 x^{\frac{5}{3}}}{5} + 3 x^{\frac{2}{3}} + C " data-equation-content=" \displaystyle F(x) = - \frac{21 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} + \frac{21 x^{\frac{5}{3}}}{5} + 3 x^{\frac{2}{3}} + C " /> </p> </p>