INTEGRATION
Substitution
Integration
byPartsFormula
l f a n i n t e g r a n hd a st h e l o r m J ' t u t . r) l u ' ( x ) . t h e nr e w r i t et h ee n t i r e
tlu : u'(r) tlx'.
integralin termsof r and its difTerenttal
I u r r ) r " ( .dr )x : u ( r ) u ( x )*
I
tt
|
u't"n') ctr
I l t u ( . r ) ) a ' ( . r l d . rI : . ft u l d , ,
JJ
TABLE
OFINTEGRALS
BasicForms
rn*r.
r. [ ,,,,u
r r:
*r.
n*l
J
t
t.
o.
l.
A.
v.
tu.
/rin
/.nr;
l
I
". I
tr.
ta.
,0.
I
I
I
,'r.
,f.
* Cl
t.
,u.
,t.
I
I
I
I
I
r tlu : tanhu* C
Isech2
I
udu:-sech u*C
Isechatanh
t.
c o t h u d u : - c s c uh * C
Icschu
sec
u tlu: ln lsecrr * tanul + C
tU.
u e ' tdu:
u
\ v , u - t \e " "+ (-
:
e " " s inbur lu
butlu :
e"" cos
,"" -
#( t r
#Qr
lhutlu:ulnu*ul_C
: I
Trigonometric
Forms
to. sin2
uttu:
c
:, )sin2u+
I
,,au:
tt.
+ j ,tn2u* C
)u
I.ur2
+('
' r nr 1
(l
Ir''
-coth u * C
csch2rrdu :
,r.
cotutlu: ln lsinul + C
u "eoudu -
s e c h u d u : t a n - l ; r i n hu l + C
tO.
tt.
(l
cotu
h t l u : l n l s i n hu l + C
, t u : t n l , " jrrnl * .
Icscha
t a nu d u : l n l s e cu l + C
cscu tltr: In lcsctr - cotul + C
tanhudu: lncoshu * C
,r.
c s c r c out d u : - c s c u * C
Forms
andLogarithmic
Exponential
,r,
I
r " r u t a n u t l u: s e c u
* C
1 6 .f + : - - : f
ll'
J u-1
I
ul_C
Icoshur/u:sinh
,0.
f5. t .lL::sin-r!+('
(t
J Juz-u2
t.
,t.
ucln:sina*C
|,"r2utlu:tanrr*C
/.r.2
u*C
Isinhudu:cosh
ndu:--cosrr*C
utlu:-c.ttr
d u : t n l l nu l * c
,0.
11'
Inu
utnu
- r]+c
* r)rnu
Forms
Hyperbolic
lr"ttu:e'+C
.l
23.[ -L
J
rn C
I +: trrt*
4 . [ ,,' d, ,: ! (
s.
r r# _ l
:
,r. u' tnu,t,
I
ffi[(r
,ttt_,u
t tr4 r,
s i nb tr- b c o sb n )* C
cosbu*b sinbu)+ C
3r.
tr.
tr.
I
I
I
t a n 2u d u : t a n r r- u * C
cotu
2 t l u : - c o t u - u* C
sin3u ctu: -
l.or3
1(2
+ sin2rz)cosu * c
, ,t, : !(2 + cos2a) sina * C
oo.
I
r ; r nu3d u :
^.
cotu
3 d u : - : c o t 2 u _ l n l s i nu l f C
or.
I
I
sec3
udu:
)ror'a
* l n l c o s u*l C
j ,". utanLt+j r" lsecu+ tanul * c
-lrin'-l
4 4 . I r i r nu t l u :
JnnJ
r.u, u +'
- |
2
[ sin" urtu
r-
f
67,
,:
Jt/ut-utdu:-lut-utl
u
f _
I.nr"-lasina
4s. /.or'udu:
JnnJ
F o r m s f n v o l vf fiin, a
g> o
ucotu+ltnlcsca-cotr.rl*c
43./.r.t utlu:-1.r.
.lntl
+'-|
u t . u 2 J ;-2u 2 o:ut e u 2 - , i t J ; 2- u 2 +f , i " , i * ,
I
/.nr'-2udu
t-l
- , r "I | - t f f l . ,
* . Jf J ; 2 _ t 1 2 .
"=du:1f,2-;
- [ t a n " - 2u t l u
4 6 . I n n nu t l u : l , r n " - t ,
n-t
J
J
-"
. o , ' - t u - [ c o t t t - 2u t l u
47. /.nt" udu :
n-t
J
J
-2
s e c u" r l u :
ur.
*lanrsec"
I
-l
f
-
s e c n ' 2t t t l u
I
tr-)
f
, c o t r . r c s c " -ut + ; ; l
c s c "u d u :
49.
u *=
;l
., .n_tI+c
1,'l:-u:
l'1 .
70.
du:-;{o'-ut-si
a
,z
J
u
:
c
t
u
u
.
[
2
t
tl
_
1t rr . I
i u - - u -_*2T, s, ,i 2
n .- .' - * C
I rttL- - l.l'
J
1/
^
csc' - utlu
11
s i n ( a- b ) u
f
50. lsinausinbutlu
J
2(u-h\
'4.I
".I
*l
*l
n c o s ud u :
'
n-l
1tr
tttJ
'l t l
sin'*l ucos--l u
n+m
t'
tsinudu
il+Dl
,r.
)
/ sin, , cor,,, u,lu
02.
f
2u}_ l
Jucos-'urlu-
u u: L J u ,- u ,- r n l , + , G 2- , 2 1 +c
f
rI
81.
d u: y l r 2- r , z ' - o " o r - r
^*,
du: -ry+rn ln+{rt-il1+c
du
n---=:
Vu- _e-
f
rI
u2du
n==:
vu"_Q-
f
8 2 .I
|
4
t
cos-'rt
u
4
ctu
_:
f
du
83. '
_-_
J ( u ' - a z ) r l/ '
- u :; +
r8 4 . L / u 2 + t r 2t l u :
J
| urtu
:
uut.
i*,
+tan-r
lutan-
.l
u2
|
7Inl
/ ill
-
+ n u :- t t z l * C
J;' -,F
-C
;,
u
_+C
u 2J u 2 - q 2
Involving
Forms
rffi
f C
I u d u : f r [ , , " * ,' i nt , , 6a.
I#]
lu'sin
t 1
rlu:
J s2Ja2_ 42
u!'l-;
/ .
ln * i u: - u-l+ C
la
r t u* "/17 *,
| uttu
:'+
sin-
a>o
..1
I,ltr,/--=--------=rr4
* u 2 t J ; 2- a 2r " + n C F- a 1 + c
f,ou2
O lu
f rit;1
J +
80.
u t lu: us i n -lu -; r/1 - u z1 g
,
-
,r.I +
,l
t : uta n -ra - j t" ,| + u 2 )+ c
u r. r an-udu
I
Iu
-t !
+ C
sln
r_
du:u.nr-|, -rf -12+C
tn.
I"or-ru
^.
8
77. I ,,) j ,2 - ,t2,lu
J
Forms
Trigonometric
Inverse
I s in- l
3ua
du
u
_-:Lf
( u . _ r z , 1 . t / z ( t 2 J u 2_ u 2
7 6 .| { ; , r :
lu'
n-l
f
.. a
+ -| s i n " - ' u c o s " 'u d u
n+nt,l
1
I
z -'
tnvofving
Forms
Jm,
tcosu,lu
'!)
u
I
8
:
n+m
tr.
J
f
t
J
C
u n s i n u- r l u "
'ilcos
la+Jaz-uzl lrl-
t u
7 4 . I u , . - u 2 ) 3 1t ]2u : - _ L * z u-, 5 o 2 v) a ' - ' t - +
in'ucos"'udu
stn'
lhl
J u 2 J u 2_ u 2
cgsudu:cosL+t/ sina*C
tt sin a du : -un ctts, * r
I
u
I
du
-:-u2ut/u--u:lC
f
73. I
l^
c o s . l r rc o sb u t l u :
;l
53. Iurinutl u:sintr-acosaI
J
du
l ,Juz-rz
s i n ( cf b ) r r
; : ;: +C
ltu+nt
s i n ( r r* b ) u
s i n ( t t- l t l u
+
+ C
)kt _ l,
,,u * ^
c o s ( c r- b ) a
c o s ( a* h ) r . r
f
52. I sinuucosbadu:- ;; , ;: +C
- rrt
/
.
t
Q
ttd + Dl
J
51.
f
l:-_
u ttlo"*u"l
,a>0
.,2
\tn(uiJu2+u2)+c
f _
n+-1
u d u: #
ur.
Iu"cos-r
t , ,+
[r'-r..,r- | #]
n+-l
* . u ' t a nu-dt :u#
I
t o n - -r , ,
I+#]
[r"'r
n+-l
85. I u2t/u2 + u2du
J
A
: ! u 2 + 2 u 2 1 { '+2u 2 - | r n l u+ J } + , 1 7+ c
8
8
*P1.,
@=,1,:,fq\u2-"h
ru.
I
I
-qt*rn(a +JF+,21
r.,.
+c
I ry du:
a
f
,1-.
8 8 .| - + :
l
l n ( ,+ / , P + ; 1 + c
'f a2 a u2
t1 tt'
-f u'
- 1, ^ | f f i
e o t. - ! ! : :
a
J uJu2+u2
er. I
J
o,
I
u
102.
*'l *,
f
tlu
92. ,
--::rC
J (uz * u:1.\/z
J \u + hu)t
-2a)t/a *bu *C
2u' Ji + lra
J uJu*bu
2nu
un-t tlu
I
6pn*, = t,er+DJm
i r r>
ro
a ,,|ffi
4l*,
Ja
lJu+bu+Jul
u 2 U /4 2 1 u 2
du
f
l' "o-5' - I - : - I u nJ u 1 b u
lu*ttul
bt2n-3t f
du
2 a t n- l l I u n - t J a + b u
,or.
lgPd,,:-+.il#
Forms
Involving
J2au -7, o > 0
ro8.
I tq;;= ou: T6;,
J
- = ! - * l , n p *hbLu l * C
fi:1a bu)
I
- ,'z*
!.o,-t (+).,
l oe. | ,' /zr, -,P a,
o',
*"|*,
e 8 I. u ( u+ b u ),' : -L t-l fu+- n u t - 1
' nlI " u I
u'
J
t /
9' ,9. .l t[u * h u t 2 -- :0 3 \ a + a u
"!L
Jr+h,
a ( n- l ) u n - t
ct,:2Jct+hu*"
| #,
t u.l ry
- 4a(a-t bu) 1za2tn1'+ bul)1g
l{o+ t u12
+
2b
.:
2
3t](bu
u
s 6 [. u L- ( uo ' , - - ' a u * 4u '' n l u *u n ' l r .
lbul
I
J
I
9 7 .[
:
t o 4 t. - : L - :
oLLl
u". :lrnl-!l*.
a
'o.u
udu
ffi
un tlu
er.f +
+ : Ih k' t b u - a t n l+ab u l1) . s
J 0+nu
J ufu*bul
-"
#t"l'to*bu13t2 I "-tJo+na'f
r u 3 f'
J Ji+n:
I
ag
*bu
F o r mIsn v o l v i n
e s I.
f
J
: _To'_j] -,
u2r/u2 y y2
s4.t [+:
.l albu
:
+ , ,- i ^ , + , c l+ , 2 1c+
E et.+ : : J , ,
,t
lo1. u"JT+ouau
"I
'!-+4
u2 - _
2 c t r n l a + l , rl t +
)c
u+nu
no. t --!L:=:
J ,/2uu-u2
f-l
- Za)(a+ buf12*,
roo. ,Ju + n a, :
;rpQbu
/
fll
f
l:
vE,,-,r* {.o,-,(i)
*.
-')
c()sr ('
+c
a
\
ttu
r5;;
/
-7
-L/-
uu
J uJ2ou - u2
ESSENTIAL
THEOREMS
ValueTheorem
lntermediate
Theorem
of Calculus,
PartI
TheFundamental
I f f ( . r ) i s c o n t i n u o u s o n a c l o s e d i n t e r v[ a l, b ] a n d f ( a ) * . l ' ( b ) ,
then for every valueM betweenf (a) and.l'(b), thereexistsat least
one valuec € (a, b) suchthat /(c) - M.
Assumethat l'(x) is continuouson [a, b] andlet F(x) be an
of /(x) on [a, b]. Then
antiderivative
Theorem
MeanValue
on
If I (x) is continuouson a closedinterval la, bl anddifl'erentiable
(a, b), then thereexistsat leastone valuec e (a, b) suchthat
rt.
J ' ( h \- . l ( u t
/ (t'l:
b_o
Interval
ona Closed
Values
Extreme
If f (x) is continuouson a closedinterval la, b], then /(x) attains
both a minimum and a maximum valueon [4, b]. Furthermore,if
c e la, bl and.f (c) is an extremevalue(min or max), then c is either
a critical point of I (.r) in (a, b) or one of the endpointsa or b.
f.o
, r r r d x : F ( b -) F ( a )
Theorem
of Calculus,
Partll
Fundamental
Assumethat /(x) is a continuousfunction on [a, b]. Then the area
of' I rxl.that is,
functionA(r) -- [- f ,,t dr isananticlerivative
Ju
A' (x) : f (x)
or equivalently +
dx
['
Ja
f tt I dt : .f (x)
: g.
Furthermore,A(x) satisfiesthe initial contlition A1,u7