X2 T04 01 integration by parts (12)
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Transcript of X2 T04 01 integration by parts (12)
![Page 1: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/1.jpg)
Integration By Parts
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Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.
![Page 3: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/3.jpg)
Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.
vduuvudv
![Page 4: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/4.jpg)
Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.
vduuvudv
u should be chosen so that differentiation makes it a simpler function.
![Page 5: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/5.jpg)
Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.
vduuvudv
u should be chosen so that differentiation makes it a simpler function.
dv should be chosen so that it can be integrated
![Page 6: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/6.jpg)
e.g. (i) xdxx cos
![Page 7: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/7.jpg)
e.g. (i) xdxx cos xu
![Page 8: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/8.jpg)
e.g. (i) xdxx cos xu
dxdu
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e.g. (i) xdxx cos xu
dxdu xdxdv cos
![Page 10: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/10.jpg)
e.g. (i) xdxx cos xu
dxdu xdxdv cos
xv sin
![Page 11: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/11.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin dxdu xdxdv cos
xv sin
![Page 12: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/12.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin dxdu xdxdv cos
xv sin
cxxx cossin
![Page 13: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/13.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log
dxdu xdxdv cos
xv sin
cxxx cossin
![Page 14: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/14.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
![Page 15: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/15.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu
![Page 16: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/16.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
![Page 17: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/17.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
xv
![Page 18: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/18.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log
dxxx log
xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
xv
![Page 19: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/19.jpg)
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log
dxxx log
xu log
cxxx log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
xv
![Page 20: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/20.jpg)
1
0
7 dxxeiii x
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1
0
7 dxxeiii x xu
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1
0
7 dxxeiii x xu
dxdu
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1
0
7 dxxeiii x xu
dxdu dxedv x7
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1
0
7 dxxeiii x xu
dxdu dxedv x7
xev 7
71
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1
0
7 dxxeiii x xu
dxdu dxedv x7
xev 7
71
1
0
71
0
7
71
71 dxexe xx
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1
0
7 dxxeiii x xu
dxdu dxedv x7
xev 7
71
1
0
71
0
7
71
71 dxexe xx
1
0
77
491
71
xx exe
![Page 27: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/27.jpg)
1
0
7 dxxeiii x xu
dxdu dxedv x7
xev 7
71
1
0
71
0
7
71
71 dxexe xx
1
0
77
491
71
xx exe
491
498
4910
491
71
7
77
e
ee
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xdxeiv x cos
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xdxeiv x cos xeu
![Page 30: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/30.jpg)
xdxeiv x cos xeu
dxedu x
![Page 31: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/31.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
![Page 32: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/32.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
![Page 33: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/33.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
![Page 34: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/34.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
![Page 35: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/35.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
dxedu x
![Page 36: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/36.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
dxedu x xdxdv sin
![Page 37: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/37.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
dxedu x xdxdv sin
xv cos
![Page 38: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/38.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
xdxexexe xxx coscossinxeu
dxedu x xdxdv sin
xv cos
![Page 39: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/39.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
xdxexexe xxx coscossin
xexexdxe xxx cossincos2
xeu
dxedu x xdxdv sin
xv cos
![Page 40: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/40.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
xdxexexe xxx coscossin
xexexdxe xxx cossincos2
xeu
dxedu x xdxdv sin
xv cos
cxexexdxe xxx cos21sin
21cos
![Page 41: X2 T04 01 integration by parts (12)](https://reader034.fdocuments.in/reader034/viewer/2022051209/54961d83b479596a4d8b4ea0/html5/thumbnails/41.jpg)
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
xdxexexe xxx coscossin
xexexdxe xxx cossincos2
xeu
dxedu x xdxdv sin
xv cos
cxexexdxe xxx cos21sin
21cos
Exercise 2B; 1 to 6 ac, 7 to 11, 12 ace etc