1. Clicker Question (Not marked) Why are there seasons on Earth? – A. The Earth’s orbit is elliptical, not circular. – B. The Sun is not at the center of the Earth’s orbit. – C. The Earth is tilted on its rotational axis. – D. It’s necessary for agriculture and food production. – E. It’s so we can water ski and snow ski in the same general location.

2. Clicker Question 1 What is ? (Hint: It’s pretty easy) – A. ((1/2)x 2 + 3x) / ((1/3)x 3 + 3x 2 + 3x) + C – B. ln(|x 2 + 6x + 3|) + C – C. 1 / 2(x 2 + 6x + 3) 2 + C – D. (1/2) ln(|x 2 + 6x + 3|) + C – E. ln(|x + 6|) + C

3. Clicker Question 2 What is ? – A. 2 ln |x – 1| + ln |x – 2| + C – B. ln |x 2 – 3x + 2| + C – C. 3 ln |x 2 – 3x + 2| + Arctan(x) + C – D. -4 ln |x – 1| + 5 ln |x – 2| + C – E. ln | (x – 1) / (x – 2)| + C

4. Antidifferentiation Strategies (9/25/13) 1. Is the integrand immediately recognizable in terms of our basic antidifferentiation facts? 2. If no, can it be algebraically manipulated into a simpler form? 3. If no, can a u-substitution be used? 4. If no, is it a product of two parts, one of which has a “nice” derivative and the other a “not too bad” antiderivative? Then try integration by parts.

5. Strategies Continued Does it have something of the form a 2  x 2 ? Then a trig substitution may be helpful. Is it a rational function? Partial fractions may work to create two or more simpler ones. All else failing, you may be able to get an answer from a Table of Integrals or from Computer Software (e.g., Mathematica). Remember that you can always do numerical (approximate) integration on a definite integral! (Next topic.)

6. Clicker Question 3 What is ? – A. (1/3) x 3 + ln(x 2 + 3) + C – B. ((1/5)x 5 + x 3 + (1/2)x 2 ) / ((1/3)x 3 + 3x) + C – C. ((1/5)x 5 + x 3 + (1/2)x 2 ) ln(x 2 + 3) + C – D. (1/3)x 3 + (1/2)ln(x 2 + 3)) + C – E. Too tired........

7. Assignment for Friday Review all the anti-derivative techniques we have learned and finish up any exercises we have done over the last two weeks which you may not have finished previously. Extra credit (3 points, due Friday): Compute in two different ways: by partial fractions (easy) and by a trig sub (not as easy, as usual). Be sure to show clearly that your two answers are the same.