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