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%\usepackage{lipsum}


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\parskip 10pt
\parindent 0pt
\def\u{\vskip -5pt}
\def\v{\vskip 10pt}
\def\s{\vskip 5pt}
\begin{document}


\begin{frame}

Integration by parts:

Derivative of a product: ~~ $(uv)' = uv' + vu'$
\v\v
\centerline{$uv' + vu'= (uv)' $}
\v\v
\centerline{$uv' = (uv)' - vu'$}
\v\v
\centerline{$\int uv' = \int (uv)' - \int vu'$}
\v\v
\centerline{$\int uv' = (uv) - \int vu'$}


\end{frame}

\begin{frame}

Integration by parts:

Derivative of a product: ~~ $(uv)' = uv' + vu'$
\v\v
\centerline{$uv' + vu'= (uv)' $}
\v\v
\centerline{$uv' = (uv)' - vu'$}
\v\v
\centerline{$\int uv' ~~~= \int (uv)'~~~ - \int vu'$}
\v\v
\centerline{$\int uv' = (uv) - \int vu'$}
\end{frame}

\begin{frame}
Example:  $\int x^4 sin(x)$\

$
\begin{array}{rclrcl}
u & = & x^4, & dv & = & sin(x)\cr
du &=& 4x^3,  &v &=& -cos(x) \cr
d^2u &=& 12x^2, &\int v &=& -sin(x)\cr
d^3u &=&24x, &\int\int v &=& cos(x)\cr
d^4u &=& 24, &\int\int\int v &=& sin(x)\cr
d^5u &=&0, &\int\int\int\int v &=& -cos(x)
\end{array}
$

 $\int x^4 sin(x)
= -$

\end{frame}


\begin{frame}
Example:  $\int x^4 cos(x)$\

$
\begin{array}{rclrcl}
u & = & x^4, & dv & = & cos(x)\cr
du &=& 4x^3,  &v &=& sin(x) \cr
d^2u &=& 12x^2, &\int v &=& -cos(x)\cr
d^3u &=&24x, &\int\int v &=& -sin(x)\cr
d^4u &=& 24, &\int\int\int v &=& cos(x)\cr
d^5u &=&0, &\int\int\int\int v &=& sin(x)
\end{array}
$

 $\int x^4 cos(x)
= x^4sin(x)
 - [4x^3(-cos(x))] 
+ [12x^2(-sin(x)]
- [24xcos(x)]
+[24sin(x)]
-\int 0dx$

\end{frame}

\begin{frame}

 $\int x^4 cos(x)dx 
= x^4sin(x)
 - \int 4x^3sin(x)dx $
\vskip 10pt
\rightline{$=  x^4sin(x)
 - [ 4x^3(-cos(x)) -   \int 12x^2(-cos(x))dx ]$}
\vskip 10pt
\rightline{$=  x^4sin(x)
 -  4x^3(-cos(x))  +   \int 12x^2(-cos(x))dx ]$}
\end{frame}



\begin{frame}


 $\int x^4 cos(x)dx
= x^4sin(x)
 - [4x^3(-cos(x))] $
\vskip 10pt
\rightline{$+ [12x^2(-sin(x)]
- [24xcos(x)]$}
\vskip 10pt
\rightline{$+[24sin(x)]
-\int 0dx$}
\vskip 15pt
\rightline{$= x^4sin(x)
 + 4x^3cos(x) - 12x^2sin(x)$}

\vskip 10pt
\rightline
{$- 24xcos(x)+24sin(x)
+C$}

 $\int x^4 cos(x)dx
= x^4sin(x)
 - [4x^3(-cos(x))] $
\vskip 10pt
\rightline{$+ [12x^2(-sin(x)]
- [24xcos(x)]$}
\vskip 10pt
\rightline{$+[24sin(x)]
+ C$}

Example:  $\int x^4 sin(x)$




Let $u = x^4$, ~$dv =sin(x)$




\end{frame}


\begin{frame}
Example:
$\int e^{2x} sin(3x)dx $
\s

Let $u = sin(3x)$, $dv = e^{2x}$

then $du = 3cos(3x)$, $v = {1 \over 2}e^{2x}$

then $d^2u = -9sin(3x)$, $\int v = {1 \over 4}e^{2x}$
\s

$\int e^{2x} sin(3x)dx $

\vskip 12pt
\rightline{$={1 \over 2}sin(3x)e^{2x} - {3 \over 4}cos(3x)e^{2x} +\int (-{9 \over 4}) sin(3x)e^{2x}dx$}

\vskip 14pt
\rightline{$={1 \over 2}sin(3x)e^{2x} - {3 \over 4}cos(3x)e^{2x} - {9 \over 4}\int  sin(3x)e^{2x}dx$~~~~}


\end{frame}


\begin{frame}

$\int e^{2x} sin(3x)dx$

\vskip 10pt
\rightline{$={1 \over 2}sin(3x)e^{2x} - {3 \over 4}cos(3x)e^{2x} - {9 \over 4}\int  e^{2x}sin(3x)dx$}
\s\s
Use algebra to solve for $\int e^{2x} sin(3x)$:
\vskip 15pt
\s
\centerline{${13 \over 4} \int e^{2x} sin(3x)dx = {1 \over 2}sin(3x)e^{2x} - {3 \over 4}cos(3x)e^{2x}$}

\s\s\s
Thus $\int e^{2x} sin(3x) dx = {4 \over 13}[{1 \over 2}sin(3x)e^{2x} - {3 \over 4}cos(3x)e^{2x}]$


\end{frame}

\end{document}

Optional Exercise:  Calculate $\int e^{x} cos(2x)$

