12 x1 t06 01 integration using substitution (2013)
TRANSCRIPT
Integration Using Substitution
Integration Using Substitution
Try to convert the integral into a “standard integral”
Integration Using Substitution
Try to convert the integral into a “standard integral”STANDARD INTEGRALS
dxxn 0 if ,0;1 ,1
1 1
nxnxn
n
Integration Using Substitution
Try to convert the integral into a “standard integral”STANDARD INTEGRALS
dxxn 0 if ,0;1 ,1
1 1
nxnxn
n
dxx1
0 ,ln xx
Integration Using Substitution
Try to convert the integral into a “standard integral”STANDARD INTEGRALS
dxxn 0 if ,0;1 ,1
1 1
nxnxn
n
dxx1
0 ,ln xx
Integration Using Substitution
Try to convert the integral into a “standard integral”STANDARD INTEGRALS
dxxn 0 if ,0;1 ,1
1 1
nxnxn
n
dxx1
0 ,ln xx
dxeax 0 ,1 ae
aax
Integration Using Substitution
Try to convert the integral into a “standard integral”STANDARD INTEGRALS
dxxn 0 if ,0;1 ,1
1 1
nxnxn
n
dxx1
0 ,ln xx
dxeax 0 ,1 ae
aax
dxax cos 0 ,sin1 aax
a
Integration Using Substitution
Try to convert the integral into a “standard integral”STANDARD INTEGRALS
dxxn 0 if ,0;1 ,1
1 1
nxnxn
n
dxx1
0 ,ln xx
dxeax 0 ,1 ae
aax
dxax cos 0 ,sin1 aax
a
dxax sin 0 ,cos1 aax
a
dxax sec2 0 ,tan1 aax
a
dxax sec2 0 ,tan1 aax
a
dxaxax tansec 0 ,sec1 aax
a
dxax sec2 0 ,tan1 aax
a
dxaxax tansec 0 ,sec1 aax
a
dxxa
122 0 ,tan1 1 a
ax
a
dxax sec2 0 ,tan1 aax
a
dxaxax tansec 0 ,sec1 aax
a
dxxa
122 0 ,tan1 1 a
ax
a
dxxa
122 axaa
ax
0, ,sin 1
dxax sec2 0 ,tan1 aax
a
dxaxax tansec 0 ,sec1 aax
a
dxxa
122 0 ,tan1 1 a
ax
a
dxxa
122 axaa
ax
0, ,sin 1
dxax
122 0 ,ln 22 axaxx
dxax sec2 0 ,tan1 aax
a
dxaxax tansec 0 ,sec1 aax
a
dxxa
122 0 ,tan1 1 a
ax
a
dxxa
122 axaa
ax
0, ,sin 1
dxax
122 0 ,ln 22 axaxx
dxax
122 22ln axx
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
duu 2 u
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
duu 2 u
2
duu
1u
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
tan xdx logsec x
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
complementary trig ratio complement of the answer
tan xdx logsec x
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
complementary trig ratio complement of the answer
e.g.cot x dx
tan xdx logsec x
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
complementary trig ratio complement of the answer
e.g.cot x dx
tan xdx logsec x
cot is the complement of tan
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
complementary trig ratio complement of the answer
e.g.cot x dx log cosec x
tan xdx logsec x
cot is the complement of tan
so the answer is minus and the
complement of sec is cosec
“NON” STANDARD INTEGRALSNot listed in the standard integrals, however will save time if you know
ln xdx ln x x x
duu 2 u
2
duu
1u
complementary trig ratio complement of the answer
e.g.cot x dx log cosec x
tan xdx logsec x
logsin xcot is the complement of tan
so the answer is minus and the
complement of sec is cosec
e.g. (i) dxxx 42
e.g. (i) dxxx 4242 xu
e.g. (i) dxxx 4242 xu
u
e.g. (i) dxxx 4242 xu
xdxdu 2u
e.g. (i) dxxx 4242 xu
xdxdu 2udu
21
e.g. (i) dxxx 4242 xu
duu
dxxx
21
2
21
4221
xdxdu 2udu
21
e.g. (i) dxxx 4242 xu
duu
dxxx
21
2
21
4221
xdxdu 2udu
21
cu 23
32
21
e.g. (i) dxxx 4242 xu
duu
dxxx
21
2
21
4221
xdxdu 2udu
21
cu 23
32
21
cu 23
31
e.g. (i) dxxx 4242 xu
duu
dxxx
21
2
21
4221
xdxdu 2udu
21
cu 23
32
21
cu 23
31
cx 23
2 431
e.g. (i) dxxx 4242 xu
duu
dxxx
21
2
21
4221
xdxdu 2udu
21
cu 23
32
21
cu 23
31
cx 23
2 431
cxx 4431 22
dx
xxxii
7841
2
dx
xxxii
7841
2
dxxduxxu
88784 2
dx
xxxii
7841
2
dxxduxxu
88784 2
udu
81
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
udu
81
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
cxx 784log81 2
udu
81
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
cxx 784log81 2
udu
81
3log xx
dxiii
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
cxx 784log81 2
udu
81
3log xx
dxiii
xdxdu
xu
log
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
cxx 784log81 2
udu
81
3log xx
dxiii
xdxdu
xu
log
duuudu
3
3
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
cxx 784log81 2
udu
81
3log xx
dxiii
xdxdu
xu
log
duuudu
3
3
cu
cu
2
2
21
21
dx
xxxii
7841
2
dxxduxxu
88784 2
cu log81
cxx 784log81 2
udu
81
3log xx
dxiii
xdxdu
xu
log
duuudu
3
3
cu
cu
2
2
21
21
c
x 2log2
1
dx
xxiv
1
dx
xxiv
1ududx
xuux2
11 2
dx
xxiv
1ududx
xuux2
11 2
udu
uu 21 2
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
uduuu 21 2
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
cxx 12132 3
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
cxx 12132 3
cxxx 121132
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxv
cxx 12132 3
cxxx 121132
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxvxdxduxu
cossin
cxx 12132 3
cxxx 121132
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxvxdxduxu
cossin
cxx 12132 3
cxxx 121132
23
3sin,
3
u
ux
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxvxdxduxu
cossin
cxx 12132 3
cxxx 121132
23
3sin,
3
u
ux
1 2
sin,2
u
ux
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxvxdxduxu
cossin
1
23
5duu
cxx 12132 3
cxxx 121132
23
3sin,
3
u
ux
1 2
sin,2
u
ux
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxvxdxduxu
cossin
1
23
5duu
123
6
61 u
cxx 12132 3
cxxx 121132
23
3sin,
3
u
ux
1 2
sin,2
u
ux
dx
xxiv
1ududx
xuux2
11 2
duu 12 2
cuu
3
312
uduuu 21 2
2
3
5 cossin
xdxxvxdxduxu
cossin
1
23
5duu
123
6
61 u
38437
231
61
66
cxx 12132 3
cxxx 121132
23
3sin,
3
u
ux
1 2
sin,2
u
ux
4
3225 x
xdxvi
4
3225 x
xdxvixdxdu
xu2
25 2
4
3225 x
xdxvixdxdu
xu2
25 2
9,416,3
uxux
4
3225 x
xdxvixdxdu
xu2
25 2
16
9
21
9
16
21
21
duu
udu
9,416,3
uxux
4
3225 x
xdxvixdxdu
xu2
25 2
16
9
21
9
16
21
21
duu
udu
16
9
21
u
9,416,3
uxux
4
3225 x
xdxvixdxdu
xu2
25 2
16
9
21
9
16
21
21
duu
udu
16
9
21
u
1916
9,416,3
uxux
5
0
225 dxxvii
5
0
225 dxxvii
ududx
xuux
cos55
sinsin5 1
5
0
225 dxxvii
ududx
xuux
cos55
sinsin5 1
2
1sin,50
0sin,0
1
1
u
uxuux
5
0
225 dxxvii
ududx
xuux
cos55
sinsin5 1
2
0
2 cos5sin2525
uduu
2
1sin,50
0sin,0
1
1
u
uxuux
5
0
225 dxxvii
ududx
xuux
cos55
sinsin5 1
2
0
2 cos5sin2525
uduu
2
0
2
2
0
2
cos25
cos5cos25
udu
uduu
2
1sin,50
0sin,0
1
1
u
uxuux
5
0
225 dxxvii
ududx
xuux
cos55
sinsin5 1
2
0
2 cos5sin2525
uduu
2
0
2
2
0
2
cos25
cos5cos25
udu
uduu
2
0
2
0
2sin21
225
2cos1225
uu
duu2
1sin,50
0sin,0
1
1
u
uxuux
5
0
225 dxxvii
ududx
xuux
cos55
sinsin5 1
2
0
2 cos5sin2525
uduu
2
0
2
2
0
2
cos25
cos5cos25
udu
uduu
2
0
2
0
2sin21
225
2cos1225
uu
duu2
1sin,50
0sin,0
1
1
u
uxuux
425
0sin21
2225
5
0
225 dxxvii
5
0
225 dxxviiy
x5
5
-5
225 xy
5
0
225 dxxviiy
x5
5
-5
225 xy
5
0
225 dxxviiy
x5
5
-5
225 xy
425
541 2
Exercise 6C; 1, 2ace, 3, 4ace, 5ace etc, 6, 7 & 8ac, 11, 12Exercise 6D; 1, 2a, 3, 4b, 5ac, 6ace etc, 7ab(i,ii)
8ab (i,iii,vi), 9ab (ii,iv,vi), 11, 13
Patel Exercise 2A ace in all NOTE: substitution is not given in Extension 2
5
0
225 dxxviiy
x5
5
-5
225 xy
425
541 2