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Section 7.1 : Integration by Parts
8. Evaluate \( \displaystyle \int{{{y^6}\cos \left( {3y} \right)\,dy}}\) .
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Hint : Doing this with “standard” integration by parts would take a fair amount of time so maybe this would be a good candidate for the “table” method of integration by parts.
Okay, with this problem doing the “standard” method of integration by parts (i.e. picking \(u\) and \(dv\) and using the formula) would take quite a bit of time. So, this looks like a good problem to use the table that we saw in the notes to shorten the process up.
Here is the table for this problem.
\[\begin{array}{rrr} {{y}^{6}} & \cos \left( 3y \right) & + \\ 6{{y}^{5}} & \displaystyle \frac{1}{3}\sin \left( 3y \right) & - \\ 30{{y}^{4}} & \displaystyle -\frac{1}{9}\cos \left( 3y \right) & + \\ 120{{y}^{3}} & \displaystyle -\frac{1}{27}\sin \left( 3y \right) & - \\ 360{{y}^{2}} & \displaystyle \frac{1}{81}\cos \left( 3y \right) & + \\ 720y & \displaystyle \frac{1}{243}\sin \left( 3y \right) & - \\ 720 & \displaystyle -\frac{1}{729}\cos \left( 3y \right) & + \\ 0 & \displaystyle -\frac{1}{2187}\sin \left( 3y \right) & - \\ \end{array}\] Show Step 2Here’s the integral for this problem,
\[\begin{align*}\int{{{y^6}\cos \left( {3y} \right)\,dy}} & = \left( {{y^6}} \right)\left( {\frac{1}{3}\sin \left( {3y} \right)} \right) - \left( {6{y^5}} \right)\left( { - \frac{1}{9}\cos \left( {3y} \right)} \right) + \left( {30{y^4}} \right)\left( { - \frac{1}{{27}}\sin \left( {3y} \right)} \right)\\ & \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {120{y^3}} \right)\left( {\frac{1}{{81}}\cos \left( {3y} \right)} \right) + \left( {360{y^2}} \right)\left( {\frac{1}{{243}}\sin \left( {3y} \right)} \right)\\ & \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {720y} \right)\left( { - \frac{1}{{729}}\cos \left( {3y} \right)} \right) + \left( {720} \right)\left( { - \frac{1}{{2187}}\sin \left( {3y} \right)} \right) + c\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*} & \frac{1}{3}{y^6}\sin \left( {3y} \right) + \frac{2}{3}{y^5}\cos \left( {3y} \right) - \frac{{10}}{9}{y^4}\sin \left( {3y} \right) - \frac{{40}}{{27}}{y^3}\cos \left( {3y} \right)\\ & \hspace{0.5in} + \frac{{40}}{{27}}{y^2}\sin \left( {3y} \right) + \frac{{80}}{{81}}y\cos \left( {3y} \right) - \frac{{80}}{{243}}\sin \left( {3y} \right) + c\end{align*}}\end{align*}\]