Clicker Question 1 What is  cos 3 (x) dx ? – A. ¼ cos 4 (x) + C – B. -3cos 2 (x) sin(x) + C – C. x – (1/3) sin 3 (x) + C – D. sin(x) – (1/3) sin 3 (x) + C – E. sin 2 (x) cos(x) + C

Clicker Question 2 What is  tan 4 (x)sec 4 (x)dx ? – A. (1/25)tan 5 (x)sec 5 (x) + C – B. tan 7 (x) + tan 5 (x) + C – C. tan 6 (x) + tan 4 (x) + C – D. (1/6) tan 6 (x) +(1/4) tan 4 (x) + C – E. (1/7)tan 7 (x) + (1/5)tan 5 (x) + C

Trig Substitutions (9/19/12) Motivation: What is the area of a circle of radius r ? Put such a circle of the coordinate system centered at the origin and write down an integral which would get the answer. Can you see how to evaluate this integral?

Replacing Algebraic With Trigonometric An expression like  (1 – x 2 ) may be difficult to work with in an integral since the inside is a binomial. Perhaps the Pythagorean Identities from trig might help since they equate a binomial (1 – cos 2 (  )) with a monomial (sin 2 (  )). So try letting x = sin(  ) and go from there. You must also replace dx !

An example What is  1 /(1 – x 2 ) 3/2 dx ? Regular substitution? Well, try a “trig sub”. Let x = sin(  ), so that dx = cos(  ) d . Now rebuild the integrand in terms of trig functions of . Can we integrate what we now have?? (Yes, think back!) Finally, we must return to x for the final answer.

Back to the circle Try a trig substitution. Since this is a definite integral, we can eliminate x once and for all and stick with  (so we must replace the endpoints also!). We’ve now proved the most famous formula in geometry.

Other trig substitutions Recall that d/dx(arctan(x)) = 1/ (1 + x 2 ) Hence  1 / (1 + x 2 ) dx = arctan(x) + C This leads to the idea that integrands which contain expressions of the form (a 2 + x 2 ) (where a is just a constant) may be related to the tan function. Thus in the expression above we make the substitution x = a tan(  ) (so dx = a sec 2 (  )d  ). What about expressions of the form (x 2 – a 2 )?

Assignment for Friday Make sure you understand and appreciate the derivation of the area of a circle. Assignment: Read Section 7.3 and do Exercises 1, 3 (see Exercise 23 in Section 7.2), 4 (The answer is 2/15. Since it’s a definite integral, eliminate x completely, including endpoints), and 9.