Summary Mathematics 144: Semester notes

substitution Rule



g Tay Algin gtx
outer inner


Jtigix gladx Jf a du
I dx Always check for sub first
of singe

J rinlinx x dx
composite th sin int
tnx inner fn
let y tnx dude Vx du lx dx
H Jsinfuldu
F costa to
H coslinx to
s


ey fo x dx Definite integral


J
let U
j
X2
xox

i d4 dx
2X
du Ox dx Multiply by 1 21
field
change boundaries i to u du values
if x p y O f it x y 1112 1
fi endu swap boundaries multiply by FD

,H G f 2
f endu
1
42 00 0 11211 40

or
J yet'd
Jo xox let 4 4 2
i du 2xdx
life du
H 112 to 1126 to du Xdx

Joxd dy
C in e x
11218 1

Scottdx Always write it o cos tan
ay
Joos sinxdx
Spin cost do
let u sink
i duldx cost du cosxdx
SH du S lady Intul to
Intul to dude insult I Yu
misinxltc

, at J 02 1 x dx
Idt U X2 1
i duldx 2x du 2Xdx
1128141112 du

H 112141312 to
312
43 02 11312 to

b J VX 1
XOX U XY
I IX 1 xox du i dx
17841112 utildu du dx

84312 tu 2 du
I t 4112th
4312th to
42 1 LTD
215 X 1 512 2 312
tc
I 3 x 1

,eg FX It V2TSX


It V2t3x
1 Let y fix Y
2 Domain 213 70
x y 43
Range y I

3 Y It V2tax 180118 For X
V2t3X
YF
Ltsx
ly1
x Y 12 2
3

4 Swap X t for f lx
y
f lx 431 112 43 Pf lx X711 Of FX1
Range



sketch

, x2 X X 12
eg Y
t complete square




y X X
X X MGM
y 4211
x 11212 V4 x 7112




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2 Dp X 7112 2 D
Rp H C ai D
y
3 1 11212 114
4
x 1142
I
Yt't
Vytha t 12 X

4 f A 112
8 44 Dr xx a Range HN
t
112 to the g
Range f lx going
y
i f lx must go to the D the sign
ye