I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 7.1 : Integration by Parts
8. Evaluate \( \displaystyle \int{{{y^6}\cos \left( {3y} \right)\,dy}}\) .
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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,
\[\begin{align*}\int{{{y^6}\cos \left( {3y} \right)\,dy}} & = \left( {{y^6}} \right)\left( {\frac{1}{3}\sin \left( {3y} \right)} \right) - \left( {6{y^5}} \right)\left( { - \frac{1}{9}\cos \left( {3y} \right)} \right) + \left( {30{y^4}} \right)\left( { - \frac{1}{{27}}\sin \left( {3y} \right)} \right)\\ & \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {120{y^3}} \right)\left( {\frac{1}{{81}}\cos \left( {3y} \right)} \right) + \left( {360{y^2}} \right)\left( {\frac{1}{{243}}\sin \left( {3y} \right)} \right)\\ & \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {720y} \right)\left( { - \frac{1}{{729}}\cos \left( {3y} \right)} \right) + \left( {720} \right)\left( { - \frac{1}{{2187}}\sin \left( {3y} \right)} \right) + c\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*} & \frac{1}{3}{y^6}\sin \left( {3y} \right) + \frac{2}{3}{y^5}\cos \left( {3y} \right) - \frac{{10}}{9}{y^4}\sin \left( {3y} \right) - \frac{{40}}{{27}}{y^3}\cos \left( {3y} \right)\\ & \hspace{0.5in} + \frac{{40}}{{27}}{y^2}\sin \left( {3y} \right) + \frac{{80}}{{81}}y\cos \left( {3y} \right) - \frac{{80}}{{243}}\sin \left( {3y} \right) + c\end{align*}}\end{align*}\]