Download 4. i_ [!!.] = vu` ~ uv`

DERIVATI¥ES AND INlrEGRAlS
Basic Differentiation Rules
I. ; r[cu]
4.
i_
:::::l
[!!.] = vu' v-~ uv'
d
S. dr [c] = 0
dr l'
d
7. dx[x] == I
8. :r[I11 IJ =
10. !!.[e"] = euu'
dr
13. ! [sin u]
16.
= (cos u)u'
:~[cot u] =
- (csc 2 u)u'
u'
r
...;1-ud
-u'
22. -d [arccot u] = - --~
r
1 + ud
19. -d [arcsin 11] =
25. : r [sinh 11]
k+
u-
d
11. -d [log" u]
r
= -(I n- a)11
14. :r[cos 11]
=-
d
17. dr[sec u]
= (sec 11 tan u)u'
ll'
r
d
23. d
- [arcsec u]
r
15. :r[tan u]
(sin u)u'
d
29. dr [sech u]
d
- u'
d
1
r
d
,,
34. - [coth- 1 11] = - - ,
dx
1 - u-
d
35. dx [sech- 1 u]
21. d
- [arctan u]
u'
= -1-+u-,
24. dx[arccsc 11]
=
d
r
d
~
...; 1 - u-
u'
= Iu I...;~
u- - 1
27. : r [tanh u]
= - (sech 11 tanh u)u'
32. d
- [cosh- 1 u]
= (sec2 11)11'
18. dx[csc 11] ""' - (esc u cot u)11'
26. :r [cosh u] = (sinh u)11'
28. : r [coth u] = - (csch2 u)u'
=
d
,,
9. d
- [In 11] ""' r
u
d
12. dx[a"] = (In a)a"ll'
*0
u
:: (11 '),
1 1
d
20. -d [arccos u] =
~
= (cosh u)11'
31. dxd [sinh- 1 u]
d
3. - [uv] = uv' + vu'
dr
d
6. dr[u"] = nll" - 1u'
2. ![11 ± v] ""' 11' ± v'
cu'
= (sech 2 u)u'
d
30. dx [csch 11] "" - (csch u coth 11 )u '
d
II'
- u'
I I ~
II ...; u- - I
,,
33. - [tanh- 1 u] = - -,
dr
1 - ud
- u'
36. -d [csch- 1 u] = I I Jf+lj2
r
11 I + u-
=~
~~- - I
-11'
= u~
1 - 11·
Basic Integration Formulas
3.
J
J
5.
Je" d11 ., e" + C
1.
7.
9.
11.
kj(11) du = k fj(11) du
du = u
2. J [/(11)
+C
J
J
J
4. fa"du = Cnla)a"
6. Jsin
cos u du
= sin 11 + C
cot u du
= !n isin ul + C
esc u du = - Jn lcsc 11
13. Jcsc-2 11 d11
=-
cot 11
+
f
© Brooks/Cole, Cengage Learning
10.
+ cot ul + C
C
12.
14.
+
I
u
, = - arctan - + C
al + u·
a
a
d11
11
du
=-
= Jf(u) du ± J g(11) du
+C
cos 11
+C
8. fran 11 du = - In Ieos 11l
15. Jcsc u cot ud11 = - esc u
17.
± g(u)] du
C
16.
18.
J
J
J
sec 11 d11
= lnlsec 11 + tan u l + C
sec 2 11 d11 = tan 11
+C
sec u tan 11 d11 = sec u
f
f
+C
+
C
d11
. 11
.j ,
, == arcsm - + C
a·- ua
d11
I
lui
.j ~
, = - arcsec- + C
11
u- - a· a
a
TrRIG0NOMETRY
Definition of the Six Trigonometric Functions
Right triangle definitions, where 0 < 0 < Tr/2.
sin 0 = opp
hyp
esc 0 = hyp
opp
d'
(_!2'
:!l.)
(_:/l
2 . 2
!)
(_il
2 •2
h
cos 0 = ~ sec 0 = yp
hyp
adj
d'
tan (} = opp cot (} = ~
adj
opp
Circular function definitions, where 0 is any angle.
.1'
r=
sin(} =
../.t2+ y2
--~-<o_._•>
(4. 4)
(4.¥)
(4. ~)
(-1.0)
g esc(} =!:..
r
il)
2
-1)2
(_il
2
_:/l.)
(_:!l
2 • 2
)'
t
X
cos(} = r
X
sec(} = !:.
X
X
)'
Reciprocal Identities
' .r = -1sm
esc x
1
csc .r = - .sm .r
sec .t = - cos .r
cosx
1
=sec x
Double -Angle Formulas
1
tanx "" - cotx
1
cotx = - tan .r
Tangent and Cotangent Identities
sin x
cos .r
tanx
=cosx
cot .r = - .smx
I + cot2 x = csc2 x
1 + tan2 x = sec 2 x
Power-Reducing Formulas
I - cos 2u
sio2 u = - - - -
2
1 + cos 2u
cos~ 11 =
2
~
I - cos 2u
tan- u = - - - 1 + cos 211
Sum-to-Product Formulas
Cofunction Identities
x) = cos x cos(f- x) = sinx
r) = cot .r
csc(f- x) = secx tan(!!.
2 - '
sec(f-
=
sin 2u 2 sin u cos 11
cos 2u = cos2 11 - sin 2 11 = 2 cos2 u - I = I - 2 sin2 u
2 tan u
tan 2u =
~
I - tan~ u
~
Pythagorean Identities
sin 2 x + cos2 .t = 1
sin(f -
(_!2' _il)
2
tan(} =- cot(} = ~
)'
(II v) cos ("- -2- v)
. u - sm
. v = .,- cos("- +- \') sm
. (II- -- ")
s•n
2
2
. u + sm
. v = 2 sm
. - +sm
2
x) =esc x cot(f - x) = tan x
Reduction Formulas
=
sin(-x)
-sinx
csc(-x) = -cscx
sec(-x) = secx
cos(-x) = cosx
tan(-x) = -tanx
cot(-x) = - cot.r
Sum and Difference Formulas
sin(u ± v) = sin u cos v ± cos u sin v
cos(u ± l') = cos u cos v + sin 11 sin v
tan 11 ± tan v
tan (u ± v) = .....;,;;;.;.;,;.;.-=-=--1 + tan u tan ''
Product-to-Sum Formulas
sin 11 sin v = k[cos(rt - v) - cos(u + v)]
1
cos 11 cos v "" 2[cos(u - v) + cos(11 + v)]
sin u cos v =
~[sin(u
+ v) + sin(u - v)]
cos 11 sin v :::::
~[sin(u
+ v) - sin(11 - v)]
© Brooks/Cole, Cengage Learning