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\begin{document}
Your friend is working on the following homework problem: $\displaystyle \int_{-1}^1 \frac{1}{1+x^4}\ dx$. 

She decides to write the integrand as 
\[ \int_{-1}^1 \frac{1}{1+x^4}\ dx = \int_{-1}^1 \frac{ \frac{1}{x^4}}{1 + \frac{1}{x^4}}\ dx \] so that she can use substitution. The rest of her work is

\begin{quote}
    Let $u = \frac{1}{x^4}$ do that $du = -\frac{4}{x^5}\ dx$. Then 
    \[ \int_{-1}^1 \frac{1}{x^4}\ dx = \int_1^1 \frac{u}{u+1} \left(-\frac{u^{-5/4}}{4}\right)\ du\] and this last integral is zero because the limits of integration are equal to each other. So $\displaystyle \int_{-1}^1 \frac{1}{1+x^4}\ dx = 0$.
\end{quote}

However, when you plot the integrand $\frac{1}{1+x^4}$, you get the following graph:

\begin{center}
    \includegraphics[width=0.6\textwidth]{mcr8.png}
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\begin{enumerate}
    \item How can you tell from the graph that something went wrong with your friend's work?
    \item Where did she make a mistake?
\end{enumerate}

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