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warmup. 1). 2). 3). Group Problem: 15 min. solution. 6.1a: Integration rules for Antiderivatives. Find the antiderivative of f(x) = 5x 4 Work backwards (from finding the derivative) The antiderivative of f(x)=F’(x) is x 5.

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warmup

1)

2)

3)




6 1a integration rules for antiderivatives

6.1a: Integrationrules for Antiderivatives


Find the antiderivative of f(x) = 5x4

Work backwards (from finding the derivative)

The antiderivative of f(x)=F’(x) is x5


In the example we just did, we know that F(x) = x2 is not the only function whose derivative is f(x) = 2x

G(x) = x2 + 2 has 2x as the derivative

H(x) = x2 – 7 has 2x as the derivative

For any real number, C, the function F(x)=x2 + C has f(x) as an antiderivative


There is a whole family of functions having 2x as an antiderivative

This family differs only by a constant


Since the functions antiderivative

G(x) = x2 F(x) = x2 + 2 H(x) = x2 – 7 differ only by a constant, the slope of the tangent line remains the same

The family of antiderivatives can be represented by F(x) + C



Integrals such as are called antiderivativedefinite integrals because we can find a definite value for the answer.

The constant always cancels when finding a definite integral, so we leave it out!


When finding indefinite integrals, we antiderivativealways include the “plus C”.

Integrals such as are called indefinite integrals because we can not find a definite value for the answer.


Finding the antiderivative
Finding the Antiderivative antiderivative

Finding the antiderivative is the reverse of finding the derivative. Therefore, the rules for derivatives leads to a rule for antiderivatives

Example:

So


Rules for antiderivatives
Rules for Antiderivatives antiderivative

Power Rule:

Ex:

Ex:

You can always check your answers by taking the derivative!


You do
You Do antiderivative

1.

2.


Rules for finding antiderivatives
Rules for Finding Antiderivatives antiderivative

Constant Multiple and Sum/Difference:


Examples
Examples antiderivative


Example
Example antiderivative


Recall
Recall antiderivative

Previous learning:

If f(x) = ex then f’(x) = ex

If f(x) = ax then f’(x) = (ln a)ax

If f(x) = ekx then f’(x) = kekx

If f(x) = akx then f’(x) = k(ln a)akx

This leads to the following formulas:



Examples1
Examples antiderivative


Indefinite integral of x 1
Indefinite Integral of x antiderivative-1

Note: if x takes on a negative value, then lnx will be undefined. The absolute value sign keeps that from happening.


Example1
Example antiderivative


5x + C antiderivative


No Calculators antiderivative


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