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6.1 Indefinite Integrals and Slope 6.1 Indefinite Integrals and Slope FieldsFields
I. The Indefinite IntegralI. The Indefinite Integral
Let Let ff be a derivative. The set of all antiderivatives be a derivative. The set of all antiderivatives
of of ff is the is the indefinite integralindefinite integral of of ff and is denoted and is denoted
by by and and where where ( )f x dx ( ) ( )f x dx F x C
( ) ( )d
F x C f xdx
II. The Indefinite Integral ContinuedII. The Indefinite Integral Continued
Is one indefinite integral of Is one indefinite integral of f, f, namely namely
the one whose value at the one whose value at a a equals 0.equals 0.
( )x
a
f t dt
III. Integral FormulasIII. Integral Formulas
Indefinite IntegralIndefinite Integral
1.) a. nx dx
1.) b. dx
x
2.) kxe dx
1
, 11
nxC n
n
ln x C
kxeC
k
Indefinite IntegralIndefinite Integral 3.) sin kx dx
4.) cos( )kx dx 25.) sec xdx
cos( )kxC
k
sin( )kxC
k
tan x C
26.) csc xdx
7.) sec tanx xdx
cot x C
sec x C
Indefinite IntegralIndefinite Integral
8.) csc cotx xdx
2
19.)
1dx
x
2
110.)
1dx
x
csc x C
arcsin x C
arctan x C
2
111.)
1dx
x x
12.) xa dx
arcsec x C
ln
xaC
a
Indefinite IntegralIndefinite Integral
13.) tan xdx
cot x x C
14.) cot xdx
ln cos x C
tan x x C
cos
sin
xdx
x
215.) tan xdx 216.) cot xdx
2sec 1x dx
2csc 1x dx
sin
cos
xdx
x
ln sec x C
ln sin x C
Indefinite IntegralIndefinite Integral
17.) sec xdx
18.) csc xdx - ln csc cotx x C
ln sec tanx x C
IV. Properties of Indefinite IntegralsIV. Properties of Indefinite Integrals
1.) Constant Multiple Rule- 1.) Constant Multiple Rule-
2.) Sum/Difference Rule- 2.) Sum/Difference Rule-
( ) ( )kf x dx k f x dx
( ) ( ) ( )f x g x dx f x dx g x dx
V. Solving Initial Value ProblemsV. Solving Initial Value Problems
Def.- Differential Equation – Any equation Def.- Differential Equation – Any equation containing a derivative.containing a derivative.
To solve a differential equation means to find a To solve a differential equation means to find a function meeting all conditions.function meeting all conditions.
Ex.- Solve Ex.- Solve
Proc:Proc:
sindy
xdx
sindy xdx
1 2cosy C x C
sindy xdx
cosy x C
If given the initial condition (0, 2), then…If given the initial condition (0, 2), then…
cos 3y x
3 C
2 cos 0 C
VI. ApplicationsVI. Applications
SPSE $1,000 is invested in an account that pays SPSE $1,000 is invested in an account that pays 6% yearly interest compounded continuously. 6% yearly interest compounded continuously. How much money will be in the account after How much money will be in the account after 25years?25years?
First, First, yy((tt) = money in the account at time ) = money in the account at time t t and and yy(0) = $1,000(0) = $1,000..
.06dy
ydt
.06dy
dty
1 2ln .06y C t C
.06dy
dty
ln .06y t C
ln .06y t Ce e
.06ty Ae
.06t Cy e e
01000 Ae
.061000 ty e
(.06)25(25) 1000 $4,481.68y e
A helicopter pilot drops a package 200 ft. above A helicopter pilot drops a package 200 ft. above ground when the helicopter is rising at a speed ground when the helicopter is rising at a speed of 20 ft./sec. How long does it take the package of 20 ft./sec. How long does it take the package to hit the ground and what is its speed on to hit the ground and what is its speed on impact?impact?
( ) 32dv
a tdt
32dv dt
( ) 32v t t C
32dv dt
(0) 20 32(0)v C
( ) 32 20ds
v t tdt
2( ) 16 20s t t t C
( ) 32 20ds
v t dt t dtdt
2(0) 200 16 0 20 0s C
2( ) 16 20 200s t t t
20 16 20 200t t
51 33 sec.
8t
5( 1 33 ) 20 33 ft./sec.8
v