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  % EDIT THIS NEXT LINE: Replace ``Name'' with your own name
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\large{\textbf{MATH 101 V01 -- ASSIGNMENT 3}}

\normalsize

\medskip

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\begin{enumerate}

% QUESTION 1
\item 

  \begin{enumerate}

  \item
  Let $a>0$.  Use properties of the integral and integration by substitution 
  to prove for any function $f$ that is continuous on the closed interval 
  $[-a,a]$:

    \begin{enumerate}

    \item
    If $f$ is an \textit{even} function ($f(-x)=f(x)$ for all $x$), then 
    \[
      \int_{-a}^a f(x)\,dx=2\int_0^a f(x)\,dx.
    \] 

    \item
    If $f$ is an \textit{odd} function ($f(-x)=-f(x)$ for all $x$), then 
    \[
      \int_{-a}^a f(x)\,dx=0.
    \]

    \end{enumerate}

  \item
  Let $L>0$ be a constant and let $n$ be a positive integer.
  Evaluate the two integrals
  \[ 
    a_n=\frac{1}{L}\int_{-L}^L x^2\cos\left(\frac{n\pi x}{L}\right)\,dx,
    \qquad
    b_n=\frac{1}{L}\int_{-L}^L x^2\sin\left(\frac{n\pi x}{L}\right)\,dx.
  \]

  \end{enumerate}

\medskip

\textit{Solution:}

% TYPE YOUR SOLUTION TO QUESTION 1 HERE

\bigskip

% QUESTION 2
\item  


  \begin{enumerate}

  \item 
  Use integration by parts to derive the 
  following \textit{reduction formula}
  \[
  \int\cos^n x\,dx=\frac{1}{n}\cos^{n-1}x\sin x+\frac{n-1}{n}\int\cos^{n-2}x\,dx,
  \]
  where $n\geq2$ is an integer.

  \item 
  Use the reduction formula to evaluate 
  \[ \int_0^{\pi/2}\cos^4x\,dx. \]

  \end{enumerate}
 
\medskip

\textit{Solution:}

% TYPE YOUR SOLUTION TO QUESTION 2 HERE  


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