អាំងតេក្រាល (បារាំង: Intégral; អង់គ្លេស: Integral) ហៅជា អនុកល [១] ក៏បាន គឺជាគន្លឹះដ៏សំខាន់នៅក្នុងគណិតវិទ្យា ។ បើនិយាយឱ្យស្រួលស្តាប់ទៅ អាំងតេក្រាល គឺជាអនុគមន៍មុនពេលធ្វើដេរីវេ ។
រូបមន្តអាំងតេក្រាលមិនកំណត់មួយចំនួន
[កែប្រែ]
C ជាចំនួនពិត
រូបមន្តអាំងតេក្រាលមិនកំនត់សំខាន់ៗ
![{\displaystyle (1).\int e^{x}\,dx\,\!=e^{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/950b78a6251e856fd94733ed0dd364d938c658b0)
![{\displaystyle (2).\int a^{x}\,dx\,\!={\frac {1}{\ lna}}~a^{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/114c6b2303202ef0f606688cae757c436031307c)
![{\displaystyle (3).\int {\frac {1}{\ x}}~dx=ln|x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd1a185f5ae478d60f9f20f9fabfa5769fb9706)
ដែល p ជាចំនួនពិត)
![{\displaystyle (5).\int {\frac {1}{\ {x^{2}-a^{2}}}}\,dx\,\!={\frac {1}{\ {2a}}}ln|{\frac {x^{2}-a^{2}}{x^{2}+a^{2}}}|+C\quad (a\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23660526d584a5804cf7fb80a70acbaa54138eec)
![{\displaystyle (6).\int {\frac {1}{\ {x^{2}+a^{2}}}}\,dx\,\!={\frac {1}{\ {a}}}tan^{-1}{\frac {x}{a}}+C\quad (a\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/605ecd8586d0083baf73a488b430ed3326c09af2)
![{\displaystyle (7).\int sinx\,dx\,\!=-cosx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93880daf64090e8931666cfe5dce936396bb1b0f)
![{\displaystyle (8).\int cosx\,dx\,\!=sinx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a0b5f6055eca78d4bfb5a720d903cd84f369c5)
![{\displaystyle (9).\int {\frac {1}{\ cos^{2}x}}\,dx\,\!=tanx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f283d6a5d82b41b74ac820dae356ad03cd634a16)
![{\displaystyle (10).\int {\frac {1}{\ sin^{2}x}}\,dx\,\!=-cotx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b96120b707a8b2c51fb3f1d0429b275471e357)
![{\displaystyle (11).\int tanx\,dx\,\!=-ln|cosx|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c85fd5185632fe3b332a9fda21ceb0c46b5c0e6)
![{\displaystyle (12).\int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,dx\,\!=sin^{-1}{\frac {x}{a}}+C\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52d68c44c0de6021f88e795d84849dd1153db0b2)
![{\displaystyle (13).\int {\frac {1}{\sqrt {x^{2}-A}}}\,dx\,\!=ln|x+{\sqrt {x^{2}+A}}|+C\quad (A\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4990651150b7deb0cab8770a81f8df1af36ebfa0)
![{\displaystyle (14).\int {\sqrt {a^{2}-x^{2}}}\,dx\,\!={\frac {1}{\ {2}}}(x{\sqrt {a^{2}-x^{2}}}+a^{2}sin^{-1}{\frac {x}{a}})+C\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ef3de93fb9f8750a95530fd45563b73cc8ab26)
![{\displaystyle (15).\int {\sqrt {x^{2}+A}}\,dx\,\!={\frac {1}{\ {2}}}(x{\sqrt {x^{2}+A}}+Aln|x+{\sqrt {x^{2}+A}}|)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be5dfb8043798eee92b73ee7631d13f4cd462c1)
|
ឧទាហរណ៏ៈគណនាអាំងតេក្រាល
![{\displaystyle (1)\int xsinx\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06e3a348f518c1d03d808cdb61f50354c7b2bb1a)
- របៀបគិត: តាង
រួចប្រើរូបមន្តអាំងតេក្រាលដោយផ្នែក គេបាន
![{\displaystyle \int xsinx\,dx\,\!=\int x(-cosx)'\,dx\,\!=x(-cosx)-\int (x)'(-cosx)\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac9b2e97b349fb15d330b3b705862b5bd524c34)
![{\displaystyle =-xcosx+\int cosx\,dx\,\!=-xcosx+sinx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f570c0e7315e7b4296ddcaf36f34697e6b7fdb4f)
![{\displaystyle (2)\int xlnx\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd38bbee628c5c289c4fea5aef0af1eb8d967d9)
តាង
គេបាន
![{\displaystyle \int xlnx\,dx\,\!=\int lnx({\frac {1}{2}}x^{2})'\,dx\,\!=lnx({\frac {1}{2}}x^{2})-\int (lnx)'({\frac {1}{2}}x^{2})\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd1ca623d4345bc7d8d2bf089d120dbbc1c8056)
![{\displaystyle ={\frac {1}{2}}x^{2}lnx-{\frac {1}{2}}\int x\,dx\,\!={\frac {1}{2}}x^{2}lnx-{\frac {1}{4}}x^{2}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c946dede5071b7ba36a78cf58098c6da0a81685d)
គេមានអនុគមន៏
គេបាន
ឧទាហរណ៏ៈគណនាអាំងតេក្រាល
![{\displaystyle (1)\int sin^{2}xcosx\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72669d88da54f9307a076c271ba6e9a2c7fb918f)
ឧទាហរណ៍ ![{\displaystyle {\frac {1}{(x+3)(x+2)(x+5)}}\,\!={\frac {A}{x+3}}\,\!+{\frac {B}{x+2}}\,\!+{\frac {C}{x+5}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ba7db2d007296f2697270a631b10298c130cc4)
តម្រូវភាគបែង រួចប្រៀបធៀបមេគុណរួមដឺក្រេនៃ ![{\displaystyle x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13b85d29899b2b2e4931388408d51f4fb086e7ec)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle A=-{\frac {1}{2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f8ff01b19f6e65c1273196a81cccaca943788f)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle B={\frac {1}{3}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7a80fec9be45fa52338a8c09d81232665afa3a)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle C={\frac {1}{6}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97999a31aa7cb532808955c5b5989d6620c62091)
ឧទាហរណ៍ ![{\displaystyle {\frac {x^{2}+1}{(x+1)(x-2)(x+7)}}\,\!={\frac {A}{x+1}}\,\!+{\frac {B}{x-2}}\,\!+{\frac {C}{x+7}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dea22bf3994265848a115251b2b520266fec55e5)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle A=-{\frac {1}{9}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b0d7492a695aa2cad96a83c234fdb5775a6cd7)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle B={\frac {5}{27}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82343fdf452ed8315b5d32ec129ac627b9c331de)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle C={\frac {25}{27}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be2e6926cd13ced0fbdfb6009d2d99eacefc2f2)
- គ/ ករណីភាគបែងមានឫសលំដាប់ខ្ពស់
ឧទាហរណ៍ ![{\displaystyle {\frac {x+2}{(x+3)^{3}(x-1)}}\,\!={\frac {A}{(x+3)^{3}}}\,\!+{\frac {B}{(x+3)^{2}}}\,\!+{\frac {C}{x+3}}\,\!+{\frac {D}{x-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da4c3243cfb2a93844f797f4d821adaf66772084)
យក
គេបាន![{\displaystyle B=-{\frac {3}{16}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41db563908ca50413bd508f335089dbb5e921b5)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle A={\frac {1}{4}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1691f432cc4972309b5db82ebd46d204330c7758)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle D={\frac {3}{64}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a54bc5cf17f320aaf79bf51fd66cb2e66faba8f)
គុណអង្គទាំងពីរនឹង
រួចយក
គេបាន ![{\displaystyle 0=C+D;C=-{\frac {3}{64}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41550dd5dc8eb7f6aa743481ce6c26be5dd7a90b)
យក
គេបាន ![{\displaystyle B=-{\frac {3}{16}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41db563908ca50413bd508f335089dbb5e921b5)
- ឃ/ ករណីភាគបែងមានឫសកុំផ្លិច
ឧទាហរណ៍ ![{\displaystyle {\frac {x+1}{(x-2)(x^{2}+1)}}\,\!={\frac {A}{x-2}}\,\!+{\frac {Bx+C}{x^{2}+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74d1a04edc33c1339dd250cb3e69a5ddd1f5226a)
គុណអង្គទាំង២ នឹង
គេបាន ![{\displaystyle A={\frac {3}{5}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/542a9a3009b8843b3fbf2ba2a3537b7d25e6a31f)
គុណអង្គទាំង២ នឹង
រួចយក ![{\displaystyle x=i\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4438e88f347f7a4cf2a216e735895ac5ec250a87)
គេបាន ![{\displaystyle B=-{\frac {3}{5}}\,\ ;\,C=-{\frac {1}{5}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c478e658a97526c6d29e45f5db1785ea74489d8b)
- ង/ ករណីភាគបែងមានឫសកុំផ្លិចលំដាប់ខ្ពស់
ឧទាហរណ៍ ![{\displaystyle {\frac {4}{x^{4}+1}}\,\!={\frac {Ax+B}{x^{2}-{\sqrt {2}}x+1}}\,\!+{\frac {Cx+D}{x^{2}+{\sqrt {2}}x+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/831fc09f9e5f24a036db1e3527396b85ae25fb30)
ដោយ
ជាអនុគមន៍គូ គេបាន
![{\displaystyle {\frac {Ax+B}{x^{2}-{\sqrt {2}}x+1}}\,\!+{\frac {Cx+D}{x^{2}+{\sqrt {2}}x+1}}\,\!={\frac {-Ax+B}{x^{2}+{\sqrt {2}}x+1}}\,\!+{\frac {-Cx+D}{x^{2}-{\sqrt {2}}x+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e3c142d4552528a21ad930651af9e8ff72ba)
គេបាន ![{\displaystyle A=-C\ ;\,B=D\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5aa43355208f8a8a3cf0f7c131bfc3aefd0d9d6)
គុណអង្គទាំង២នឹង
រួចយក
គេបាន ![{\displaystyle A=-{\sqrt {2}}\,\ ;\,C={\sqrt {2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b344f4655159daed4684ad000e60446c97d0b6e3)
យក
គេបាន
វិធីសាស្រ្ត OSTROGRADSKI
[កែប្រែ]
ប្រើសម្រាប់គណនាអាំងតេក្រាលអនុគមន៍ប្រភាគសនិទានដែលភាគបែងមានឫសលំដាប់ខ្ពស់ ។
- បើ
មានឫសលំដាប់ខ្ពស់ច្រើន គេបាន៖
![{\displaystyle \color {blue}\int {\frac {P(x)}{Q(x)}}\,dx\,\!={\frac {X(x)}{R(x)}}\,\!+\int {\frac {Y(x)}{S(x)}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/139855d04a507091532affbe09b8ce1b1635cf60)
ដែល ![{\displaystyle R(x)=PGCD[Q(x);Q^{'}(x)]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3416fe3f88696ab8ff0a82734219e4fdf8e683cd)
![{\displaystyle S(x)={\frac {Q(x)}{R(x)}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ff0e29e32283e3e637be3d9796cf0b0e1cf39c)
និង
ជាពហុធាមានមេគុណត្រូវកំណត់ហើយមានដឺក្រេរៀងគ្នា តូចជាង
និង
មួយឯកតា
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {1}{(x^{3}-1)^{2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72875c7c34e6821a7a9218b0962c6d703f008613)
![{\displaystyle {\frac {1}{(x^{3}-1)^{2}}}\,\!={\frac {1}{(x-1)^{2}(x^{2}+x+1)^{2}}}\,\!={\frac {A}{(x-1)^{2}}}\,\!+{\frac {B}{x-1}}\,\!+{\frac {Cx+D}{(x^{2}+x+1)^{2}}}\,\!+{\frac {Ex+F}{x^{2}+x+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ad7aba9d72da17543726bd9e0a7943d22545be)
គេបាន ![{\displaystyle Q(x)=(x^{3}-1)^{2};Q^{'}(x)=6x^{2}(x^{3}-1);R(x)=PGCD[Q(x);Q^{'}(x)]=x^{3}-1;S(x)={\frac {Q(x)}{R(x)}}\,\!={\frac {(x^{3}-1)^{2}}{x^{3}-1}}\,\!=x^{3}-1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/838371b2b09eec49e6bf00bd1c815d84510e3b43)
![{\displaystyle \Rightarrow I=\int {\frac {1}{(x^{3}-1)^{2}}}\,dx\,\!={\frac {ax^{2}+bx+c}{x^{3}-1}}\,\!+\int {\frac {a^{'}x^{2}+b^{'}x+c^{'}}{x^{3}-1}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aea82e13ed38d12defd1c08be5b4b456658611a8)
ដេរីវេអង្គទាំង២ គេបាន
![{\displaystyle {\frac {1}{(x^{3}-1)^{2}}}\,\!={\frac {(2ax+b)(x^{3}-1)-3x^{2}(ax^{2}+bx+c)}{(x^{3}-1)^{2}}}\,\!+{\frac {(a^{'}x^{2}+b^{'}x+c^{'})(x^{3}-1)}{(x^{3}-1)^{2}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f8aacdd582e68bdd048c566cc7a800e3c5df90)
តម្រូវភាគបែង រួចប្រៀបធៀបមេគុណរួមដឺក្រេនៃ
គេបាន ![{\displaystyle a=0;b=-{\frac {1}{3}}\,\!;c=0;a^{'}=0;b^{'}=0;c^{'}=-{\frac {2}{3}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ab4d930dba080c6e024e584ca1fde9570a0be9)
![{\displaystyle \Rightarrow I=\int {\frac {1}{(x^{3}-1)^{2}}}\,dx\,\!={\frac {-{\frac {1}{3}}\,\!x}{x^{3}-1}}\,\!+\int {\frac {-{\frac {2}{3}}\,\!}{x^{3}-1}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f02e41a3d9e6294a9b5404abd332cb91269d8daf)
អាំងតេក្រាលអនុគមន៍អសនិទាន
[កែប្រែ]
![{\displaystyle \color {blue}I=\int f(x^{\frac {m}{n}}\,\!;x^{\frac {p}{q}}\,\!;......)\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88a631e8f3fdd82cdade8cb7da61a3fd12563765)
គេត្រូវតាង
ដែល
ជាភាគបែងរួមនៃប្រភាគ ![{\displaystyle {\frac {m}{n}}\,\!;{\frac {p}{q}}\,\!;......\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71769879f9c2414a3e3dd9535c6afc1700e17ed0)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {\sqrt[{3}]{x}}{{\sqrt {x}}\,\!+{\sqrt[{4}]{x}}}}\,dx\,\!=\int {\frac {x^{\frac {1}{3}}}{x^{\frac {1}{2}}\,\!+x^{\frac {1}{4}}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da1806eef0d8f90910508ddd94159a3075fb667)
តាង ![{\displaystyle x=t^{12}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948e137eb0eff263623f382c3e5e6d26503d7868)
![{\displaystyle \color {blue}I=\int f\left[\left({\frac {ax+b}{cx+d}}\,\!\right)^{\frac {m}{n}}\,\!;\left({\frac {ax+b}{cx+d}}\,\!\right)^{\frac {p}{q}}\,\!;......\right]\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb6f95e5a676f98b4787bbc758421b240199ada)
គេតាង
ដែល
ជាភាគបែងរួមនៃប្រភាគ ![{\displaystyle {\frac {m}{n}}\,\!;{\frac {p}{q}}\,\!;......\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71769879f9c2414a3e3dd9535c6afc1700e17ed0)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\sqrt {\frac {1-x}{1+x}}}\,dx\,\!=\int \left({\frac {1-x}{1+x}}\right)^{\frac {1}{2}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/191173cc392911b2364d086d197c150eccf0d1f0)
តាង ![{\displaystyle {\frac {1-x}{1+x}}\,=\,t^{2}\,\!\Leftrightarrow \,x={\frac {1-t^{2}}{1+t^{2}}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a951826800acd98323fef1b4ddb0d934b6dcf2e7)
វិធីសាស្រ្តប្តូរអថេរEULER
[កែប្រែ]
សម្រាប់អាំងតេក្រាលមានរាង
![{\displaystyle \color {blue}I=\int f\left(x;{\sqrt {ax^{2}+bx+c}}\right)\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d445a11e0f491c5bb742ac9d7d954ef1bfa1c0e)
- ក/ បើ Δ<0 ; a>0 តាង
![{\displaystyle {\sqrt {ax^{2}+bx+c}}\,=\,{\sqrt {a}}x+t\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc1c4969e2a020ccb0f74c932ea1ab4ed36d19e4)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {1}{x{\sqrt {x^{2}-x+3}}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b40b143355504b83c9e7eef8e058079bb2eef526)
តាង ![{\displaystyle {\sqrt {x^{2}-x+3}}\,=\,x+t\,\Rightarrow x={\frac {3-t^{2}}{1+2t}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3925e74469a9938bc178ecbdd4dbaaacc1f71eae)
- ខ/ បើ Δ<0 ; c >0 តាង
![{\displaystyle {\sqrt {ax^{2}+bx+c}}\,=\,xt+{\sqrt {c}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bce915e3cbff47ec45f91be786f13ba4daea087)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {(1-{\sqrt {1+x+x^{2}}})^{2}}{x^{2}{\sqrt {1+x+x^{2}}}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d88a3050cec95d37244cd3f50d6260846fd3863d)
តាង ![{\displaystyle {\sqrt {1+x+x^{2}}}\,=\,xt+1\Rightarrow x={\frac {1-2t}{t^{2}-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a3a1b4c306263b93ccb36a6d3dfcfe381f69e1)
- គ/ បើ Δ>0 គេបាន
![{\displaystyle {\sqrt {ax^{2}+bx+c}}\,=\,{\sqrt {a(x-x_{1})(x-x_{2})}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948bf928f2ae1c268e1ec16f5399318f9ce817fc)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {\sqrt {x^{2}+2x}}{x}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd5dfdcd59f9447662b3a32ca574dbe45073f35b)
តាង ![{\displaystyle {\sqrt {x^{2}+2x}}\,=\,xt\Rightarrow x={\frac {2}{t^{2}-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf86ea915fbc835b9438157a83646befe9d205de)
អាំងតេក្រាលរាង![{\displaystyle \color {blue}I=\int {\frac {P_{n}(x)}{\sqrt {ax^{2}+bx+c}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4854449848720952b1e9d6877c8eccf40a2f3ae)
[កែប្រែ]
គេបំលែង
![{\displaystyle \color {Red}I=\int {\frac {P_{n}(x)}{\sqrt {ax^{2}+bx+c}}}\,dx\,=\,P_{n-1}(x){\sqrt {ax^{2}+bx+c}}\,+\,\lambda \int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d315693cdc52aa00074499c4782dd4e875577731)
ជាពហុធាដឺក្រេ
មានមេគុណត្រូវកំណត់ ហើយគេអាចគណនាមេគុណទាំងនោះ ដោយដេរីវេអង្គទាំងពីរ រួចប្រៀបធៀមេគុណរួមដឺក្ររេនៃះ
។
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {x^{3}+2x^{2}+3x+4}{\sqrt {x^{2}+2x+2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6524fc50587ad8a60ff67d2baa5d156c041a1d1)
គេបាន :
![{\displaystyle I=\int {\frac {x^{3}+2x^{2}+3x+4}{\sqrt {x^{2}+2x+2}}}\,dx\,=\,(ax^{2}+bx+c){\sqrt {x^{2}+2x+2}}\,+\,\lambda \int {\frac {1}{\sqrt {x^{2}+2x+2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c91d7d1bfd46a50c8e5bccd9661d1d8eb21cbcd5)
អាំងតេក្រាលអនុគមន៍ទ្វេធាឌីផេរ៉ង់ស្យែល
[កែប្រែ]
![{\displaystyle \color {blue}I=\int x^{m}(a+bx^{n})^{p}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4acc474d0815e0c0190d99a4dc09e31f3415d5a)
គេអាចគណនាតាមបីករណី៖
បើ ![{\displaystyle p\in \mathbb {Z} \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97ce54a71bd931db23cfd01bc90b1272de5ba257)
តាង
ដែល
ជាភាគបែងរួមនៃប្រភាគ ![{\displaystyle m;n\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe7ce201be9dbcd3d4565c3b471c6961727d162)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {1}{{\sqrt {x}}({\sqrt[{4}]{x}}+1)^{10}}}\,dx\,=\,\int x^{-{\frac {1}{2}}}(x^{\frac {1}{4}}+1)^{-10}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73bd94c103d98311cc164b624b1c529ddd3d16df)
តាង ![{\displaystyle x=t^{4}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa6128b09b95c3af675e02ef31a2d5d8a1e56eb)
បើ
តាង
ជាភាគបែងរួមនៃ![{\displaystyle p\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6afd5e7bfef4bded62b9da338eaffaa90d24846b)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {\sqrt[{3}]{1+{\sqrt[{4}]{x}}}}{\sqrt {x}}}\,dx\,=\,\int x^{-{\frac {1}{2}}}(1+x^{\frac {1}{4}})^{\frac {1}{3}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fadc8b8b585a99a183aee37c15ab13a60726fd7)
តាង ![{\displaystyle 1+x^{\frac {1}{4}}=t^{3}\,\Leftrightarrow x=(t^{3}-1)^{4}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00c76d9cba6de73c6e25911797566cde22f3a38)
បើ
តាង
ឬ
ដែល
ជាភាគបែងរួមនៃ![{\displaystyle p\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6afd5e7bfef4bded62b9da338eaffaa90d24846b)
ឧទាហរណ៍ : គណនា ![{\displaystyle \int {\frac {1}{x^{4}{\sqrt {1+x^{2}}}}}\,dx\,=\,\int x^{-4}(1+x^{2})^{-{\frac {1}{2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2cd335076a89ecca6cb1aec90d9ea2cb9af96fb)
តាង ![{\displaystyle 1+x^{-2}=t^{2}\,\Leftrightarrow x=(t^{2}-1)^{-{\frac {1}{2}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8a532dd7d37ae58822a6350e9e21f1d641626c)
អាំងតេក្រាលដោយផ្នែកដែលមាន៤រាង
[កែប្រែ]
ប្រើរូបមន្តអាំងតេក្រាលដោយផ្នែក ![{\displaystyle \int udv\,=\,uv\,-\,\int vdu\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/103c6f20c1b1e1b73238337f6627627e2e97b711)
- ១/ រាង
![{\displaystyle \color {blue}\int P(x)sinaxdx\,;\,\int P(x)cosaxdx\,;\,\int P(x)e^{ax}dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0425505a5172912bd7818391fdc7e94c4c835ea)
ដែល
ជាពហុធា
ជាចំនួនថេរ គេតាង ![{\displaystyle u=P(x)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3cea2a8a85e86c0586d33c09f175bcc526e45e9)
ឧទាហរណ៍ : គណនា
តាង ![{\displaystyle u=x\Rightarrow du=dx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66a53163e084da76f6df7eaa400559a03a58ebd2)
- ២/ រាង
តាង ![{\displaystyle u=log_{a}x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/063a03882823e54759ba1c3698155a3031efbdbb)
ឧទាហរណ៍ : គណនា
តាង ![{\displaystyle u=log_{2}x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb26def2ac7a5df63801b9e5f8dfdfbe84d9e564)
- ៣/ រាង
![{\displaystyle \color {blue}\int e^{ax}sinaxdx\,;\,\int e^{ax}cosaxdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd49a3e278f737ab1067231d5dfe2b2be28d26f)
តាង ![{\displaystyle u=e^{ax}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3ee9feb6e62b4118597d4a88836d2a9cd346cd)
ឧទាហរណ៍ : គណនា
តាង ![{\displaystyle u=e^{x}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8d6ba61d06b97f4a1ab59e6381461b0cb88100)
- ៤/ រាង
តាង ![{\displaystyle u=arcsinx\,;\,u=arccosx\,;u=arctanx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/329438304c1a63f9d59c5a3d2fcad79e17829f71)
ឧទាហរណ៍ : គណនា
តាង ![{\displaystyle u=arctanx\Rightarrow \,du={\frac {1}{1+x^{2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4b4ed367b5f501c5af97ff992142542560a44c)
- ៥/ រាង
![{\displaystyle \color {blue}\int cosmxcosnxdx\,;\,\int sinmxsinnxdx\,;\,\int sinmxcosnxdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c21db71ee7bfeae526b3789639b71b93dcf4b25)
ប្រើរូបមន្ត នូឌុប
![{\displaystyle cosacosb\,=\,{\frac {1}{2}}[cos(a+b)+cos(a-b)]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42069e6b80978d2d38e26ece6efd79c2488755af)
![{\displaystyle sinasinb\,=\,{\frac {1}{2}}[cos(a-b)-cos(a+b)]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c650be027dc09953ecba649f9a89892df15782d)
![{\displaystyle sinacosb\,=\,{\frac {1}{2}}[sin(a+b)+sin(a-b)]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8dc6189958338b0a2e8a2f58bc4596bff8e165e)
ឧទាហរណ៍ : គណនា ![{\displaystyle \int sin4xsin3xdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/611f9b9596807689cf3046908bb66de67d072373)
អាំងតេក្រាលរាង ![{\displaystyle \color {blue}I=\int sin^{m}xcos^{n}xdx\,;\,(m;n\in \mathbb {N} ^{*})\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93f1e9ad2e0a9b465b51d1bb1a3a5ce781d1c9d0)
[កែប្រែ]
- ១/ បើ
សេស តាង ![{\displaystyle t=cosx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4eadada3528e5716ded2dd21caed559c1961fa)
- ២/ បើ
សេស តាង ![{\displaystyle t=sinx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735d12a6d565fa9b3781ac3eacd02a5607149867)
- ៣/ បើ
គូ ប្រើវិធីបន្ថយដឺក្រេ ![{\displaystyle cos^{2}x={\frac {1+cos2x}{2}}\,;\,sin^{2}x={\frac {1-cos2x}{2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e37bc710143b79399929b0fb89ba403e6a1981)
ឧទាហរណ៍ : គណនា
តាង ![{\displaystyle t=cosx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4eadada3528e5716ded2dd21caed559c1961fa)
អាំងតេក្រាលរាង![{\displaystyle \color {blue}I=\int sin^{m}xcos^{n}xdx\,;\,(m<0;n<0)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92f1713542c96994709ce828fa7d90ca7ea1e79c)
[កែប្រែ]
គេតាង ![{\displaystyle t=tanx\Rightarrow \,dt=(1+tan^{2}x)dx\,\Leftrightarrow dx={\frac {1}{1+t^{2}}}\,dt\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76c6a150bb5689889eef71fb7492fad6478ea6b9)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int sin^{-{\frac {3}{2}}}xcos^{-1}xdx\,=\,\int {\frac {1}{sin^{\frac {3}{2}}cosx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc42dd7bf95d4d9a7ede06bf8a1611b44506111)
បំលែង ![{\displaystyle {\frac {1}{sin^{{\frac {3}{2}}{cosx}}}}\,=\,(1+{\frac {1}{tan^{2}x}})^{\frac {3}{4}}(1+tan^{2})^{\frac {1}{2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611ade812867819f04e47d3b6f161e66a8350fa)
តាង
គេបាន ![{\displaystyle I=\int t^{-{\frac {3}{2}}}(1+t^{2})^{\frac {1}{4}}\,dt\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce290cef96d3f9532fccda4e09d38f813f765b89)
តាង ![{\displaystyle 1+t^{-2}=k^{4}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f349c113e6372942ea528837f797e0045cc2a5f)
អាំងតេក្រាលរាង![{\displaystyle \color {blue}I=\int cos^{m}xdx\,;\,\int sin^{n}xdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c91c1f6deeddd719d6c1b6bbc784f972177657f9)
[កែប្រែ]
- បើ
សេស
រៀង
សេស
ចូរប្រើរូបមន្ត ![{\displaystyle cos^{2}x+sin^{2}x=1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55a7dcb0b48614af042c63fa718f60b10964bcd5)
- បើ
គូ
រៀង
គូ
ចូរប្រើរូបមន្ត ![{\displaystyle cos^{2}x={\frac {1+cos2x}{2}}\ ;\,sin^{2}x={\frac {1-cos2x}{2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd1fecbb6ddfe5d7fc74b12103bd1dd2b4757c2)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int cos^{5}xdx\,=\,\int cos^{4}xcosxdx\,=\,\int (1-sin^{2}x)^{2}cosxdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5325d78b7f6b825b20f8ff59f6e3851ba8da0431)
តាង ![{\displaystyle t=sinx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735d12a6d565fa9b3781ac3eacd02a5607149867)
អាំងតេក្រាលរាង![{\displaystyle \color {blue}I=\int tan^{m}xdx\ ;\,J=\int cot^{n}xdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3074964173517d893d78d8e91811286d1728e18)
[កែប្រែ]
គេប្រើវីធីបន្ថយដឺក្រេ
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int tan^{5}xdx\,=\,\int tan^{3}xtan^{2}xdx\,=\,\int [tan^{3}x(tan^{2}x+1)-tan^{3}x]dx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30020e8aa8cd0d17eb892d8a0c7846d7de616360)
អាំងតេក្រាលអនុគមន៍ត្រីកោណមាត្រ ![{\displaystyle \color {blue}I=\int R(sinx\,;\,cosx)\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b353a63b461c161ec0f326c5491a78f517403d3e)
[កែប្រែ]
ជាទូទៅ គេតាង![{\displaystyle t=tan{\frac {x}{2}}\Rightarrow x=2arctant\Rightarrow dx={\frac {2}{1+t^{2}}}dt\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5556c6c39feba994fca315e6ab2cbc9352ba03)
![{\displaystyle sinx={\frac {2t}{1+t^{2}}}\,;\,cosx={\frac {1-t^{2}}{1+t^{2}}}\,;\,tanx={\frac {2t}{1-t^{2}}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd48930ca87c8285935d1e937a0107fe372a7d15)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {1}{1+sinx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e80c9ed8d876a7d4bca5c3f04d5653fea80db53)
តាង ![{\displaystyle t=tan{\frac {x}{2}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f34afd08674e33f9c1acf63924b6dd9d4b2d168)
- ក/ បើ
តាង ![{\displaystyle t=cosx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4eadada3528e5716ded2dd21caed559c1961fa)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {sin^{3}x}{cosx+sin^{2}x+1}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6954e98191aa3c94934e6a90b31e0d1c62e92d50)
![{\displaystyle t=cosx\Rightarrow dt=-sinxdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25356a718fb3ef6f72c4845bf1e8acbae7d64bfc)
- ខ/ បើ
តាង ![{\displaystyle t=sinx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735d12a6d565fa9b3781ac3eacd02a5607149867)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {cos^{5}x}{sin^{3}x+cos^{2}x-sinx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e072dc0430ff9de2f5067f410357fb19dc7a9d8)
![{\displaystyle t=sinx\Rightarrow dt=cosxdx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9afce90323e0ea56b4f416479dd89e7cdc2dbb)
- គ/ បើ
តាង ![{\displaystyle t=tanx\Rightarrow x=arctant\Rightarrow dx={\frac {1}{1+t^{2}}}dt\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c693033e43cda3a7a698d41a0c886fd95a54f028)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {cos^{2}x}{sin^{2}x+4sinxcosx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20b1b4f546d6e7220f1a27bfbd84c5ec65d886fe)
តាង ![{\displaystyle t=tanx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b5f5114f2a6b5864e2f332373e368ea361cfcc)
វិធីប្តូរអថេរត្រីកោណមាត្រ
[កែប្រែ]
- ក/ បើអនុគមន៍ក្រោមសញ្ញាអាំងតេក្រាលមានរ៉ាឌីកាល់
គេត្រូវ តាង
ឬ ![{\displaystyle x=asint\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36612fbc669af817f85f580b7b7dbe35cbec4fa0)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\sqrt {2-x^{2}}}\,dx\,=\,\int {\sqrt {{\sqrt {2^{2}}}-x^{2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ef41844ecff44a03d8fe0d38d0df9f5bb26838)
តាង ![{\displaystyle x={\sqrt {2}}sint\Rightarrow dx={\sqrt {2}}costdt\,;\,t=arcsin{\frac {x}{\sqrt {2}}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4c2fb4e6199ce122b853322e8db9b0c40157b5)
- ខ/ បើអនុគមន៍ក្រោមសញ្ញាអាំងតេក្រាលមានរ៉ាឌីកាល់
គេត្រូវតាង
ឬ ![{\displaystyle x={\frac {a}{sint}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfaebcf2a5ec2bafeaaf7487893483f5a64bfbec)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {x^{3}}{\sqrt {x^{2}-4}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9c843ad97259fb262796eb5494c30f4785b284)
តាង ![{\displaystyle x={\frac {2}{cost}}\Rightarrow dx={\frac {2sint}{cos^{2}t}}dt\,;\ t=arccos{\frac {2}{x}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19f7d4d72c08b6917b50628c13c4babda15c131d)
អាំងតេក្រាលរាង ![{\displaystyle \color {Blue}I=\int {\frac {a^{'}sinx+b^{'}cosx}{asinx+bcosx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d47257108372ff89eb5d4527390771fed74ea26d)
[កែប្រែ]
គេត្រូវបំលែង :
![{\displaystyle \color {Red}a^{'}sinx+b^{'}cosx=A(asinx+bcosx)+B(acosx-bsinx)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40496150c4cce57ff1048789032b0789d685bea2)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {sinx-cosx}{sinx+2cosx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6733892742173537dfa4b4d9f173d87a18ed6789)
ដោយ ![{\displaystyle sinx-cosx=A(sinx+2cosx)+B(cosx-2sinx)=(A-2B)sinx+(2A+B)cosx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcbcbae319e89ddb136b96fd544e2bc5eaeaa9d7)
![{\displaystyle \Rightarrow A=-{\frac {1}{5}}\,;\,B=-{\frac {3}{5}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62b4e7309e772152f3e87ceeaa02d79f30c2a174)
អាំងតេក្រាលរាង ![{\displaystyle \color {Blue}I=\int {\frac {a^{'}sinx+b^{'}cosx}{(asinx+bcosx)^{2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd47e7b9b9c2e4a1822cd0e1106977f7613cb73c)
[កែប្រែ]
គេត្រូវបំលែង
![{\displaystyle \color {Red}a^{'}sinx+b^{'}cosx=A(asinx+bcosx)+B(acosx-bsinx)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40496150c4cce57ff1048789032b0789d685bea2)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {sinx-cosx}{(2sinx+cosx)^{2}}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57029803b9eb355969949e681e20717e36d44a37)
ដោយ ![{\displaystyle sinx-cosx=A(2sinx+cosx)+B(2cosx-sinx)=(2A-B)sinx+(A+2B)cosx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49cbc0e08cbf3d7469867c4b527f0c46aa3a8dbb)
គេបាន ![{\displaystyle A={\frac {1}{5}}\,;\,B=-{\frac {3}{5}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef62dd86660e7708c3fb1c26aace93361730da7)
អាំងតេក្រាលរាង ![{\displaystyle \color {Blue}I=\int {\frac {a^{'}sinx+b^{'}cosx+c^{'}}{asinx+bcosx+c}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03988eae611da6ad690dbc4c0a281cfcb889a71)
[កែប្រែ]
គេត្រូវបំលែង
![{\displaystyle \color {Red}a^{'}sinx+b^{'}cosx+c^{'}\,=\,A(asinx+bcosx+c)+B(acosx-bsinx)+C\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b798548c8f5666fa8010da0b78eecb746a80883c)
ឧទាហរណ៍ : គណនា ![{\displaystyle \color {Blue}I=\int {\frac {sinx-2cosx+3}{sinx+2cosx-3}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6564cf2f104d7639ecd00387fe25977058699368)
ដោយ ![{\displaystyle sinx-2cosx+3\,=\,A(sinx+2cosx-3)+B(cosx-2sinx)+C\,=\,(A-2B)sinx+(2A+B)cosx-3A+C\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fcef36e78687e5b9e7d7518cfb342195a1c7b30)
គេបាន ![{\displaystyle A=-{\frac {3}{5}}\,;\,B=-{\frac {4}{5}}\,;\,C={\frac {6}{5}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed1a8f3f5978e836a5e27d51794a04384db94a5)
អាំតេក្រាលរាង![{\displaystyle \color {blue}I=\int {\frac {a^{'}sin^{2}x+2b^{'}sinxcosx+c^{'}cos^{2}x}{asinx+bcosx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2351117d27e3b322c46c0512201a0fa010b6af)
[កែប្រែ]
គេត្រូវបំលែង
![{\displaystyle \color {Red}a^{'}sin^{2}x+2b^{'}sinxcosx+c^{'}cos^{2}x\,=\,(asinx+bcosx)(Asinx+Bcosx)+C(sin^{2}x+cos^{2}x)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084a0cb4e23fba1566c451a5cc850f3c0c764499)
ឧទាហរណ៍ : គណនា ![{\displaystyle I=\int {\frac {sin^{2}x-2sinxcosx+3cos^{2}x}{sinx-cosx}}\,dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547ed516352bcdfc39b384690b0f469069c55aa9)
ដោយ ![{\displaystyle sin^{2}x-2sinxcosx+3cos^{2}x\,=\,(sinx-cosx)(Asinx+Bcosx)+C(sin^{2}x+cos^{2}x)\,=\,(A+C)sin^{2}x+(B-A)sinxcosx+(C-B)cos^{2}x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8fae70e34db378be79422cecbc323674c4bdfb)
គេបាន ![{\displaystyle A=0\,;\,B=-2\,;\,C=1\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c611ef1ae8076c8729bbaa8f2acf736d429c415)
- ↑ Viray An. (1998) The Orkida Dictionary Of English-Cambodia Language.