4.9 Antiderivatives Tues Dec 1 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 try Find the original function

Finding f(x) from f’’(x) When given a 2 nd 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 #