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\def\d{\mathrm{d}}
\def\C{\mathbb{C}}
\def\N{\mathbb{N}}
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\title{Calculus and Elements of Linear Algebra I}
\author{Homework 7}
\date{Due on Moodle, Monday, October 26, 2020}
\maketitle


\begin{enumerate}

\item Use substitution to evaluate the following integrals.
\begin{enumerate}
\item $\displaystyle \int \frac{\cos (\pi/x)}{x^2} \, \d x$
\item $\displaystyle \int_1^{\mathrm e} \frac{\ln x}{x} \, \d x$
\end{enumerate}

\item Use integration by parts to evaluate the following integrals.
\begin{enumerate}
\item $\displaystyle \int \sin (x) \, \ln (\cos x) \, \d x$
\item $\displaystyle \int_0^{\pi/2} x \, \sin (x) \, \cos (x) \, \d x$
\end{enumerate}


\item
\begin{enumerate}

\item Prove the reduction formula
\[
  \int \cos^n(x) \, \d x
  = \frac{1}{n} \, \cos^{n-1}(x) \, \sin(x)
  + \frac{n-1}{n} \int \cos^{n-2}(x) \, \d x \,.
\]
\emph{Hint:} Use integration by parts and the fact that $\cos^2(x) +
\sin^2(x) = 1$.
\item Evaluate $\displaystyle \int \cos^2(x)\,\d x$.
\item Evaluate $\displaystyle \int_0^{2\pi} \cos^4(x) \, \d x$.
\end{enumerate}


\item Find the area between the curves $x=1-y^2$ and
$y=-x-1$.



\item Suppose that $f \colon \R \to \R$ continuous and \emph{odd},
i.e., satisfies $-f(x) = f(-x)$.  Show that
\[
  \int_{-a}^a f(x) \, \d x = 0 \,.
\]


\end{enumerate}

\end{document}
