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Clicker Question 1 What is the unique antiderivative of f (x ) = 1 / x 2 whose value is 4 when x = 1 ? A. -1 /x + 5 B. -1 /x + 4 C. -1 /x + 3 D. ln(x 2) + 4 E. 1 /x + 3

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**Techniques For Finding Antiderivatives (4/22/09)**

Finding derivatives involves facts and rules ; it is a completely mechanical process. Finding antiderivatives is not completely mechanical. It involves some facts, a couple of rules, and then various techniques which may or may not work out. There are many functions (e.g., f (x) = ex^2) which have no known antiderivative formula.

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**There Are a Couple of “Rules”**

Sum and Difference Rule: Antiderivatives can be found working term by term (just like derivatives). Constant Multiplier Rule: Constant multipliers just get carried along as you get antiderivatives (just like derivatives). HOWEVER, there is no Product Rule, Quotient Rule, or Chain Rule for Antiderivatives!

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**Algebraic Manipulation**

Always look for algebraic simplification or manipulation first. Example: x 4 (3x + 7) dx Example: (4 + x 2) / x 2 dx

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**Clicker Question 2 What is (x + 2)(x + 3)dx ?**

A. (1/2 x 2 + 2x )(1/2 x 2 + 3x ) + C B. 1/3 x 3 + 6x + C C. 1 D. 1/3 x 3 + 5/2 x 2 + 6x + C E. 2x + 5

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**Reversing the Chain Rule: “substitution” or “guess and check”**

Any ideas about x 2(x 3 + 4)5 dx ?? How about x ex^2 dx ? Try ln(x) / x dx But we’ve been lucky! Try sin(x 2) dx

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**The Substitution Technique**

It’s called a “technique”, not a “rule”, because it may or may not work (our text calls it a “rule”, but I disagree!) If there is a chunk, try calling the chunk u. Compute du = (du/dx) dx Replace all parts of the original expression with things involving u (i.e., eliminate x). If you were lucky/clever, the new expression can be anti-differentiated easily.

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**Clicker Question 3 What is t sin(t 2 + 4) dt ?**

A. ½ t 2 cos(1/3 t 3 + 4t ) + C B. ½ cos(t 2 + 4) + C C. – ½ cos(t 2 + 4) + C D. – cos(t 2 + 4) + C E. ½ t 2 cos(t 2 + 4) + C

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Assignment for Friday Read Section 5.5 of the text and go over today’s class notes. In Section 5.5, do Exercises 1-15 odd and 19. Hand-in #4 will go out Friday and will be due Tuesday (4/28) at 4:45 pm.

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