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warmup

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warmup

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Group Problem: 15 min

solution

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

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

Notice there are no lower or upper limits

Integrals such as are called definite 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 always 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 is the reverse of finding the derivative. Therefore, the rules for derivatives leads to a rule for antiderivatives

Example:

So

Power Rule:

Ex:

Ex:

You can always check your answers by taking the derivative!

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Constant Multiple and Sum/Difference:

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:

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

5x + C

No Calculators