

\magnification 2200
\parindent 0pt
\parskip 10pt
\pageno=1
\hsize 7truein
\hoffset -0.35truein
\vsize 9.2truein
\def\u{\vskip -5pt}
\def\v{\vskip 10pt}



Integration by parts:

Derivative of a product:  $(uv)' = uv' + vu'$

\centerline{$uv' = (uv)' - vu'$}

\centerline{$\int uv' = \int (uv)' - \int vu'$}

\centerline{$\int uv' = (uv) - \int vu'$}


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

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

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

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

$\int e^{2x} sin(3x) = {1 \over 2}sin(3x)e^{2x} - \int {3 \over 2}e^{2x}cos(3x)$

\hskip 0.4in $= {1 \over 2}sin(3x)e^{2x} - [{3 \over 4}cos(3x)e^{2x} - \int {-9 \over 4}sin(3x)e^{2x}$


$\int e^{2x} sin(3x) = {1 \over 2}sin(3x)e^{2x} - {3 \over 4}cos(3x)e^{2x} - {9 \over 4}\int sin(3x)e^{2x}$


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


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

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

\end 
