Write the derivative Write the derivative for each of the for each of the following.following.

3

3

3

3

11. ( )

31

2. ( ) 231

3. ( ) 531

4. ( ) 23

F x x

F x x

F x x

F x x

CalculusIndefinite Integrals

Friday, April 21, 2023

(with a hint of the definite integral)

Each of the previous Each of the previous are Antiderivatives of are Antiderivatives of f(x)=xf(x)=x22

2 31( )

3f x dx x dx x C

In mathematics notation:

Integral sign

integrand VariableOf Integration

antiderivative ConstantOf Integration

Recall the Slopefield Recall the Slopefield conceptsconcepts The slopefield of the differential The slopefield of the differential

equation y’ = xequation y’ = x2 2 has infinite has infinite solutions solutions

We say the integral of f(x) with We say the integral of f(x) with respect to x has as its respect to x has as its antiderivative F(x) + Cantiderivative F(x) + C

Note that the antiderivative is the Note that the antiderivative is the inverse of derivative just as inverse of derivative just as integrationintegration is the inverse of is the inverse of differentiationdifferentiation

( ) ( )f x dx F x C

The indefinite integral The indefinite integral vs.vs.The definite integralThe definite integral

4

22

x dx

x dx

2 4

0

3 2

22

x dx

x dx

Note:Note:

The definite integral is a The definite integral is a number.number.

The indefinite integral is The indefinite integral is a family of functionsa family of functions

IntegrateIntegrate

7dx Reverse the derivative rule of a linear function

Reverse the power Reverse the power rule of differentationrule of differentation

1

1

nn xx dx C

n

Differentiation & Differentiation & Integration RulesIntegration Rules Differentiation p. 173Differentiation p. 173 Integration p. 180Integration p. 180

IntegrateIntegrate

4

22

x dx

x dx

Rewrite integrals to fit the Rewrite integrals to fit the rulesrules

3

2

2

2 2

2

11.

(2 )

2. 3

13.

4. sec

sin5.

1 sin

dxx

x dx

xdx

x

d

xdxx

Homework – Practice Homework – Practice improves your improves your understandingunderstanding P. 181 1 – 10 P. 181 1 – 10 P. 182 31 – 41 oddP. 182 31 – 41 odd P. 186 51 – 59 oddP. 186 51 – 59 odd