\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus

Please login to create an exam or a quiz.

Calculus
Anti-Derivatives
New Random

Find the anti-derivative of \(\displaystyle f(x) = \frac{- 5 x^{3} + 5 x^{2} - 2 x + 8}{\sqrt[4]{x}}\)

Using termwise division gives \(\displaystyle f(x) = - 5 x^{\frac{11}{4}} + 5 x^{\frac{7}{4}} - 2 x^{\frac{3}{4}} + \frac{8}{\sqrt[4]{x}}\). Finding the antiderivative of each term gives \(\displaystyle F(x) = - \frac{4 x^{\frac{15}{4}}}{3} + \frac{20 x^{\frac{11}{4}}}{11} - \frac{8 x^{\frac{7}{4}}}{7} + \frac{32 x^{\frac{3}{4}}}{3} + C\)

Download \(\LaTeX\) 

\begin{question}Find the anti-derivative of $f(x) = \frac{- 5 x^{3} + 5 x^{2} - 2 x + 8}{\sqrt[4]{x}}$
    \soln{9cm}{Using termwise division gives $f(x) = - 5 x^{\frac{11}{4}} + 5 x^{\frac{7}{4}} - 2 x^{\frac{3}{4}} + \frac{8}{\sqrt[4]{x}}$. Finding the antiderivative of each term gives $F(x) = - \frac{4 x^{\frac{15}{4}}}{3} + \frac{20 x^{\frac{11}{4}}}{11} - \frac{8 x^{\frac{7}{4}}}{7} + \frac{32 x^{\frac{3}{4}}}{3} + C$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}
HTML for Canvas 
<p> <p>Find the anti-derivative of  <img class="equation_image" title=" \displaystyle f(x) = \frac{- 5 x^{3} + 5 x^{2} - 2 x + 8}{\sqrt[4]{x}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7B-%205%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20-%202%20x%20%2B%208%7D%7B%5Csqrt%5B4%5D%7Bx%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = \frac{- 5 x^{3} + 5 x^{2} - 2 x + 8}{\sqrt[4]{x}} " data-equation-content=" \displaystyle f(x) = \frac{- 5 x^{3} + 5 x^{2} - 2 x + 8}{\sqrt[4]{x}} " /> </p> </p>
HTML for Canvas
<p> <p>Using termwise division gives  <img class="equation_image" title=" \displaystyle f(x) = - 5 x^{\frac{11}{4}} + 5 x^{\frac{7}{4}} - 2 x^{\frac{3}{4}} + \frac{8}{\sqrt[4]{x}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%205%20x%5E%7B%5Cfrac%7B11%7D%7B4%7D%7D%20%2B%205%20x%5E%7B%5Cfrac%7B7%7D%7B4%7D%7D%20-%202%20x%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%20%2B%20%5Cfrac%7B8%7D%7B%5Csqrt%5B4%5D%7Bx%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = - 5 x^{\frac{11}{4}} + 5 x^{\frac{7}{4}} - 2 x^{\frac{3}{4}} + \frac{8}{\sqrt[4]{x}} " data-equation-content=" \displaystyle f(x) = - 5 x^{\frac{11}{4}} + 5 x^{\frac{7}{4}} - 2 x^{\frac{3}{4}} + \frac{8}{\sqrt[4]{x}} " /> . Finding the antiderivative of each term gives  <img class="equation_image" title=" \displaystyle F(x) = - \frac{4 x^{\frac{15}{4}}}{3} + \frac{20 x^{\frac{11}{4}}}{11} - \frac{8 x^{\frac{7}{4}}}{7} + \frac{32 x^{\frac{3}{4}}}{3} + C " src="/equation_images/%20%5Cdisplaystyle%20F%28x%29%20%3D%20-%20%5Cfrac%7B4%20x%5E%7B%5Cfrac%7B15%7D%7B4%7D%7D%7D%7B3%7D%20%2B%20%5Cfrac%7B20%20x%5E%7B%5Cfrac%7B11%7D%7B4%7D%7D%7D%7B11%7D%20-%20%5Cfrac%7B8%20x%5E%7B%5Cfrac%7B7%7D%7B4%7D%7D%7D%7B7%7D%20%2B%20%5Cfrac%7B32%20x%5E%7B%5Cfrac%7B3%7D%7B4%7D%7D%7D%7B3%7D%20%2B%20C%20" alt="LaTeX:  \displaystyle F(x) = - \frac{4 x^{\frac{15}{4}}}{3} + \frac{20 x^{\frac{11}{4}}}{11} - \frac{8 x^{\frac{7}{4}}}{7} + \frac{32 x^{\frac{3}{4}}}{3} + C " data-equation-content=" \displaystyle F(x) = - \frac{4 x^{\frac{15}{4}}}{3} + \frac{20 x^{\frac{11}{4}}}{11} - \frac{8 x^{\frac{7}{4}}}{7} + \frac{32 x^{\frac{3}{4}}}{3} + C " /> </p> </p>