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Lecture notes

integration class notes + solved examples (pt1)
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This document contains basic integration rules and methods including trigonometric, hyperbolic and inverse functions, u substitution, integration by parts (tabular method), reduction formula and much more. This document is a summary for the first half of the calculus II course with solved examples ...

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  • February 26, 2023
  • 1
  • 2021/2022
  • Lecture notes
  • Ghada
  • 1-4

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indefinite integration (+C) The fundamental theorem of calculus
1- Constant → Variable
St → Xtc find y'
2- variable raised tothepower KINETIC
gp → X" + C y= f' (E) dt
4
3- e raised to the power of a variable → Se"→qI+c
ax y •= fcacxja.CH]- [fabio
get → Eftc d
differentiation y'= Fca Cx)) aka
of power.
4- one over a variable → f#→ Inxtc find y'
5- integration of a root → fax →fin → *At" y= faint
+, +C Izz dt
frt ax → fix+3¥ → 3*37+0 y' = 24ftp.e/-2yncDco)
6- f foxifcadx → fait'+ C ya 2¥
nti

7- f FEY dx → In (fax))to
Area under the curve
8- U substitution * draw the graphs
[Xsin(x3dx fÑsincu)&#z * determine where the req
area lies.
U=×3 ↳ Ssinadu
du=3Xdx * determine whether to use
⇒ COS (a) to or horizontal strip
¥2=dx ⇒ cosciac
* try to avoid having to div
into two by choosing the a
Trigonometric Functions
1- fsinxdx → -cost y IC * Verticle strip A-(fa
2- fcosxdx→ sin ✗ + c derivative
the ang. * horizontal
l Strip A=f
3- Stan x dx → f-§%#dx → - Incosx → in#→ insectic
4- f secxdx → f secx. sexttanx
seex + tan a☑ →f seex + sextan x ax → In/secattanx/to
CsCx +COEX Secx + tan x
5- fcscxda-f-cscxcscxtcotxdx-sf-CSEX-C.SCXCOM
(sax + Cox → - In/sext Coty +C

6- fseexdx → tanate

7- f CSex dx → -coexto Identities

8- f cot ✗ dx → f cos
sin ✗
x dx → In/sinx/to Sink +0054=1
sink ⇒(1- cos2x)
cost =L a + cosy;]
a-/secxtanx ax → secxie
10-f cot ✗ Cscxdx → -CSCAC Seck-tank ⇒
cosh2- sinkx =,
huperbolic Factions
Sin (a) sin (b) = Ic
1) Isin hxdx → coshXtc
cosca) cos (b) = {
2) f cos hxdx → sinhXtc
Sin (a)Coscb) =#
3) J tan hxdx → f sinhxdx → In/coshx/to
coshx

4) J sech xdx → f ax → fetze → Jefe. → f&¥e. → f 2¥, → 2f¥#d×

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