11.1 Antiderivatives and Indefinite Integrals

A. Let’s figure out the formula.B. Those with rational or negative

exponentsC. Those that need expandingD. Those that need simplifying

E. Word Problems

A. Let’s figure out the formula.

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xxxxxx

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x

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++

→+

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First add one to its exponent. Then, divide the term by its new exponent. Last, add the + C.

This is sort of undoing differentiation!

• It’s like we are answering the riddle: “What is the function whose derivative is this?”

∫

∫

++

=

+==

+

Cn

xdxx

Cxxdxx

nn

1

4 of integral indefinite" 4

1

433

∫ ∫ dxxdxx 32 12 :You try 6

( )

Cxxxx

Cxxxx

dx

dxdxxdxxdxx

dxxxx

++−+=

++−+

+−+

+−+

∫

∫∫ ∫∫∫

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3

4

4

6

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346

346

235

235

∫

+−+− dx

xxxx 14

3

223 :You try

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B. Those with rational or negative exponents

• As you have seen before, sometimes it is necessary to REWRITE the problem in exponential form without negative exponents BEFORE using the rules.

• Examples of rewriting:

∫∫ ∫

+⋅

−

34 3

4 32

6

3 3

4x

dxdx

xxdx

xx

∫∫ ∫

+⋅

−

43 2

3 22

2

4 7

6

:You try

u

dudx

xxdt

tt

C. Those that need expanding

• You know that you can integrate term-by-term. Sometimes you have to EXPAND things (F.O.I.L., distribute, etc.) before you can do the term-by-term thing.

• “Terms” are only connected to each other by plus signs. Nothing else.

( ) ( )∫ ∫ −+ dxxdxx 33 1 :You try 2

( ) ( )∫ ∫ −+ dxxxdxxx 322 1 :You try 2

∫∫+−+++−

dxx

xxxxdx

x

xxxx 645 :You try

1464 245234

D. Those that need simplifying

( ) ( ) ( ) Cxx

dxxdxx

xx ++=+=+

++∫ ∫ 3

23

5

53 2

•FACTOR TOP AND BOTTOM

•CANCEL VERTICALLY IF POSSIBLE

•INTEGRATE TERM BY TERM

dxx

xx∫ −

−−1

23 :You try

2

E. Word Problems

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The marginal cost for producing x units of a product is modeled by 32 0.04 .

It costs $50 to produce one unit. Find the total cost of producing 200 units.

dCx

dx= −