This instruction material adopted of Calculus by Frank Ayres Jr
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It should be noted that the first set of parametric equations represents only a portion of the parabola, whereas the second represents the entire curve.
THE FIRST DERIVATIVE
dx dy
is given by
du dx
du dy
dx dy
The second derivative
2 2
dx y
d
is given by
dx du
dx dy
du d
dx y
d
2 2
SOLVED PROBLEMS
1. Find
dx dy
and
2 2
dx y
d
, given x = - sin , y = 1- cos
sin ,
cos 1
d dy
d dx
, and
cos 1
sin d
dx d
dy dx
dy
2 2
2 2
cos 1
1 cos
1 1
cos 1
1 cos
cos 1
sin dx
d d
d dx
y d
12. FUNDAMENTAL INTEGRATION FORMULAS
IF Fx IS A FUNCTION Whose derivative F`x=fx on a certain interval of the x-
axis, then Fx is called an anti-derivative or indefinite integral of fx. The indefinite integral of a given function is not unique; for example, x
2
, x
2
+ 5, x
2
– 4 are indefinite integral of fx = 2x since
x x
dx d
x dx
d x
dx d
2 4
5
2 2
2
. All indefinite integrals of fx = 2x are then included in x
2
+ C where C, called the constant of integration, is an arbitrary constant.
The symbol dx
x f
is used to indicate that the indefinite integral of fx is to be found. Thus we write
C x
dx x
2
2
FUNDAMENTAL INTEGRATION FORMULAS. A number of the formulas below
follow immediately from the standard differentiation formulas of earlier chapters while Formula 25, for example, may be checked by showing that
2 2
2 2
1 2
2 2
1
arcsin u
a C
a u
a u
a u
du d
Absolute value signs appear in certain of the formulas. For example, we write 5.
C u
du d
ln
This instruction material adopted of Calculus by Frank Ayres Jr
19
instead of 5a
, ln
u C
u u
du
5b.
, ln
u C
u u
du
and 10.
C u
du u
sec ln
tan instead of
10a C
u du
u sec
ln tan
, all u such that sec u 1
10b C
u du
u sec
ln tan
, all u such that sec u -1
Fundamental Integration Formulas
1.
C x
f dx
x f
dx d
18.
C u
csc du
u cot
u csc
2.
dx v
dx u
dx v
u
19.
C a
u tan
arc a
1 u
a du
2 2
3.
dx u
a dx
au
, a any constant 20.
C a
u sin
arc u
a du
2 2
4.
1 m
, C
1 m
u du
u
1 m
m
5.
C u
ln u
du
6.
1 a
, a
, C
a ln
a du
a
u u
7.
, C
e du
e
u u
8.
C u
cos du
u sin
9.
C u
sin du
u cos
10.
C u
sec ln
du u
tan
11.
C u
sin ln
du u
cot
12.
C u
tan u
sec ln
du u
sec
13.
C u
cot u
csc ln
du u
csc
14.
C u
tan du
u sec
2
15.
C u
cot du
u csc
2
16.
C u
sec du
u tan
u sec
This instruction material adopted of Calculus by Frank Ayres Jr
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21.
C a
u sec
arc a
1 a
u u
du
2 2
22.
C a
u a
u ln
a 2
1 a
u du
2 2
23.
C u
a u
a ln
a 2
1 u
a du
2 2
24.
C a
u u
ln a
u du
2 2
2 2
25.
C a
u u
ln a
u du
2 2
2 2
26.
C a
u arcsin
a u
a u
du u
a
2 2
1 2
2 2
1 2
2
27.
C a
u u
ln a
a u
u du
a u
2 2
2 2
1 2
2 2
1 2
2
28.
C a
u u
ln a
a u
u du
a u
2 2
2 2
1 2
2 2
1 2
2
13. INTEGRATION BY PARTS
INTEGRATION BY PARTS. When u and v are differentiable function of
d uv = u dv + v du u dv = duv
– vdu i
du v
uv dv
u To use i in effecting a required integration, the given integral must be
separated into two parts, one part being u and the other part, together with dx, being dv. For this reason, integration by the use of i is called integration by parts. Two
general rules can be stated: a
the part selected as dv must be readily integrable b
du v
must not be more complex than dv
u
Example 1: Find
dx e
x
2
x 3
Take u = x
2
and dx
x e
dv
x
2
; then du = 2x dx and
2
2 1
x
e v
. Now by the rule C
e e
x dx
e x
e x
dx e
x
x x
x x
x
2 2
2 2
2
2 1
2 2
1 2
2 1
2
This instruction material adopted of Calculus by Frank Ayres Jr
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Example 2: Find
dx x
2 ln
2
Take u = ln x
2
+ 2 and dv = dx; then
2 2
2
x dx
x du
and v = x. By the rule.
dx x
x x
x dx
x x
x dx
x 2
4 2
2 ln
2 2
2 ln
2 ln
2 2
2 2
2 2
C x
arc x
x x
2 tan
2 2
2 2
ln
2
REDUCTION FORMULAS. The labour involved in successive applications of
integration by parts to evaluate an integral may be materially reduced by the use of reduction formulas. In general, a reduction formula yields a new integral of the same
form as the original but with an exponent increased or reduced. A reduction formula succeeds if ultimately it produces an integral which can be evaluated. Among the
reduction formulas are: A.
1 ,
2 2
3 2
2 2
1
1 2
2 1
2 2
2 2
2
m u
a du
m m
u a
m u
a u
a du
m m
m
B.
2 1
, 1
2 2
1 2
1 2
2 2
2 2
2 2
m du
u a
m ma
m u
a u
u d
u a
m m
m
C. 1
, 2
2 3
2 2
2 1
1 2
2 1
2 2
2 2
2
m a
u du
m m
a u
m u
a a
u du
m m
m
D.
2 1
, 1
2 2
1 2
1 2
2 2
2 2
2 2
m du
a u
m ma
m a
u u
u d
a u
m m
m
E.
du e
u a
m e
u u
d e
u
au m
au m
au m
1
2 1
F.
du u
m m
m u
u u
d u
m m
m 2
1
sin 1
cos sin
sin
G.
du u
m m
m u
u u
d u
m m
m 2
1
cos 1
sin cos
cos
H.
du u
u n
m n
n m
u u
u d
u u
n m
n m
m m
2 1
1
cos sin
1 cos
sin cos
sin n
m du
u u
n m
m n
m u
u
n m
n m
, cos
sin 1
cos sin
2 1
1
I.
du bu
u b
m bu
b u
u d
bu n
m m
m
cos cos
sin
1
J.
du bu
u b
m bu
b u
u d
bu n
m m
m
sin sin
cos
1
This instruction material adopted of Calculus by Frank Ayres Jr
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14. TRIGONOMETRIC INTEGRALS
