7.1-bo'lim : Qismlar bo'yicha integratsiya
8. \( \displaystyle \int{{{y^6}\cos \left( {3y} \right)\,dy}}\) ni baholang.
Maslahat: Buni qismlar bo'yicha "standart" integratsiya bilan bajarish juda ko'p vaqtni oladi, shuning uchun bu qismlar bo'yicha integratsiyaning "jadval" usuli uchun yaxshi nomzod bo'lishi mumkin.
Yechim
Xo'sh, bu muammo bilan qismlar bo'yicha integratsiyaning "standart" usulini bajaring (i.e. \(u\) va \(dv\) ni tanlash va formuladan foydalanish) biroz vaqt talab etadi. Shunday qilib, bu jarayonni qisqartirish uchun eslatmalarda ko'rgan jadvaldan foydalanish yaxshi muammoga o'xshaydi.
Mana bu muammo uchun jadval.
\[\begin{array}{rrr} {{y}^{6}} & \cos \left( 3y \right) & + \\ 6{{y}^{5}} & \displaystyle \frac{1}{3}\sin \left( 3y \right) & - \\ 30{{y}^{4}} & \displaystyle -\frac{1}{9}\cos \left( 3y \right) & + \\ 120{{y}^{3}} & \displaystyle -\frac{1}{27}\sin \left( 3y \right) & - \\ 360{{y}^{2}} & \displaystyle \frac{1}{81}\cos \left( 3y \right) & + \\ 720y & \displaystyle \frac{1}{243}\sin \left( 3y \right) & - \\ 720 & \displaystyle -\frac{1}{729}\cos \left( 3y \right) & + \\ 0 & \displaystyle -\frac{1}{2187}\sin \left( 3y \right) & - \\ \end{array}\]2-qadam
Mana bu muammoning ajralmas qismi,
\[\begin{align*}\int{{{y^6}\cos \left( {3y} \right)\,dy}} & = \left( {{y^6}} \right)\left( {\frac{1}{3}\sin \left( {3y} \right)} \right) - \left( {6{y^5}} \right)\left( { - \frac{1}{9}\cos \left( {3y} \right)} \right) + \left( {30{y^4}} \right)\left( { - \frac{1}{{27}}\sin \left( {3y} \right)} \right)\\ & \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {120{y^3}} \right)\left( {\frac{1}{{81}}\cos \left( {3y} \right)} \right) + \left( {360{y^2}} \right)\left( {\frac{1}{{243}}\sin \left( {3y} \right)} \right)\\ & \,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {720y} \right)\left( { - \frac{1}{{729}}\cos \left( {3y} \right)} \right) + \left( {720} \right)\left( { - \frac{1}{{2187}}\sin \left( {3y} \right)} \right) + c\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*} & \frac{1}{3}{y^6}\sin \left( {3y} \right) + \frac{2}{3}{y^5}\cos \left( {3y} \right) - \frac{{10}}{9}{y^4}\sin \left( {3y} \right) - \frac{{40}}{{27}}{y^3}\cos \left( {3y} \right)\\ & \hspace{0.5in} + \frac{{40}}{{27}}{y^2}\sin \left( {3y} \right) + \frac{{80}}{{81}}y\cos \left( {3y} \right) - \frac{{80}}{{243}}\sin \left( {3y} \right) + c\end{align*}}\end{align*}\]