x2 t04 01 by parts (2013)
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Integration By Parts
Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.
Integration By PartsWhen an integral is a product of two functions and neither is the derivative of the other, we integrate by parts.
vduuvudv
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.
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
e.g. (i) xdxx cos
e.g. (i) xdxx cos xu
e.g. (i) xdxx cos xu
dxdu
e.g. (i) xdxx cos xu
dxdu xdxdv cos
e.g. (i) xdxx cos xu
dxdu xdxdv cos
xv sin
e.g. (i) xdxx cos xu
xdxxx sinsin dxdu xdxdv cos
xv sin
e.g. (i) xdxx cos xu
xdxxx sinsin dxdu xdxdv cos
xv sin
cxxx cossin
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log
dxdu xdxdv cos
xv sin
cxxx cossin
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
xv
e.g. (i) xdxx cos xu
xdxxx sinsin
xdxii log
dxxx log
xu log
dxdu xdxdv cos
xv sin
cxxx cossin
xdxdu dxdv
xv
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
1
0
7 dxxeiii x
1
0
7 dxxeiii x xu
1
0
7 dxxeiii x xu
dxdu
1
0
7 dxxeiii x xu
dxdu dxedv x7
1
0
7 dxxeiii x xu
dxdu dxedv x7
xev 7
71
1
0
7 dxxeiii x xu
dxdu dxedv x7
xev 7
71
1
0
71
0
7
71
71 dxexe xx
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
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
xdxeiv x cos
xdxeiv x cos xeu
xdxeiv x cos xeu
dxedu x
xdxeiv x cos xeu
dxedu x xdxdv cos
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
dxedu x
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
dxedu x xdxdv sin
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsinxeu
dxedu x xdxdv sin
xv cos
xdxeiv x cos xeu
dxedu x xdxdv cos
xv sin
xdxexe xx sinsin
xdxexexe xxx coscossinxeu
dxedu x xdxdv sin
xv cos
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
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
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
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