4.9 AntiderivativesWed Jan 7

Do Now

If f ’(x) = x^2, find f(x)

Antiderivatives• Antiderivative - the original function in a

derivative problem (backwards)

• F(x) is called an antiderivative of f(x) if F’(x) = f(x)

• g(x) is an antiderivative of f(x) if g’(x) = f(x)

• Antiderivatives are also known as integrals

Integrals + C

• When differentiating, constants go away

• When integrating, we must take into consideration the constant that went away

Indefinite Integral

• Let F(x) be any antiderivative of f. The indefinite integral of f(x) (with respect to x) is defined by

where C is an arbitrary constant

Examples

• Examples 1.2 and 1.3

The Power Rule

• For any rational power

• 1) Exponent goes up by 1• 2) Divide by new exponent

Examples

• Examples 1.4, 1.5, and 1.6

The integral of a Sum

• You can break up an integrals into the sum of its parts and bring out any constants

You try

Trigonometric Integrals• These are the trig integrals we will work

with:

Exponential and Natural Log Integrals

• You need to know these 2:

Example

• Ex 1.8

Integrals of the form f(ax)

• We have now seen the basic integrals and rules we’ve been working with

• What if there’s more than just an x inside the function? Like sin 2x?

Integrals of Functions of the Form f(ax)

• If , then for any constant ,

• Step 1: Integrate using any rule• Step 2: Divide by a

Note

• While this works for “basic” chain rule functions, it does not work for anything more than a linear ‘inside’

Examples

• Ex 1.9

Extra Do Now if needed – ignore this

• Do Now• Integrate• 1)

• 2)

Revisiting the + C

• Recall that every time we integrate a function, we need to include + C

• Why?

Solving for C

• We can solve for C if we are given an initial value.

• Step 1: Integrate with a + C• Step 2: Substitute the initial x,y values• Step 3: Solve for C• Step 4: Substitute for C in answer

Examples

You tryFind the original function

Finding f(x) from f’’(x)

• When given a 2nd derivative, use both initial values to find C each time you integrate

• EX: f’’(x) = x^3 – 2x, f’(1) = 0, f(0) = 0

Acceleration, Velocity, and Position

• Recall: How are acceleration, velocity and position related to each other?

Integrals and Acceleration

• We integrate the acceleration function once to get the velocity function– Twice to get the position function.

• Initial values are necessary in these types of problems

Closure

• Find the original function f(x)

• HW: p.280 #1-71 odds

4-9 Anti-derivativesThurs Jan 8

• Do Now• Integrate and find C• 1)

• 2)

HW Review

Closure

• Journal Entry: What is integration? How are integrals and derivatives related? Why do we include +C?

• HW: Ch 4 Ap questions (AP4-1) All of them