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Clicker Question 1

Clicker Question 1. What is cos 3 ( x ) d x ? 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 )d x ? A. (1/25)tan 5 ( x )sec 5 ( x ) + C

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Clicker Question 1

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  1. Clicker Question 1 • What is cos3(x) dx ? • A. ¼ cos4(x) + C • B. -3cos2(x) sin(x) + C • C. x – (1/3) sin3(x) + C • D. sin(x) – (1/3) sin3(x) + C • E. sin2(x) cos(x) + C

  2. Clicker Question 2 • What is tan4(x)sec4(x)dx ? • A. (1/25)tan5(x)sec5(x) + C • B. tan7(x) + tan5(x) + C • C. tan6(x) + tan4(x) + C • D. (1/6) tan6(x) +(1/4) tan4(x) + C • E. (1/7)tan7(x) + (1/5)tan5(x) + C

  3. Trig Substitutions (9/18/13) • 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?

  4. Replacing Algebraic With Trigonometric • An expression like (1 – x2) 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 – sin2()) with a monomial (cos2()). • So try letting x = sin() and go from there. You must also replace dx !

  5. An example • What is 1 /(1 – x2)3/2 dx ? Regular substitution? • Well, try a “trig sub”. Let x = sin() , so thatdx = 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.

  6. Clicker Question 3 • What is x / (1 – x2)5 dx? • A. –1 / (8( 1 – x2)4) + C • B. 1 / (6( 1 – x2)6) + C • C. 1 / (12( 1 – x2)6) + C • D. 1 / (8( 1 – x2)4) + C • E. –1 / (10( 1 – x2)5) + C

  7. 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.

  8. Other trig substitutions • Recall that d/dx(arctan(x)) = 1/ (1 + x2)Hence 1 / (1 + x2) dx = arctan(x) + C • This leads to the idea that integrands which contain expressions of the form (a2 + x2) (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 sec2()d). • What about expressions of the form (x2 – a2)?

  9. 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. • Hand-in #1 is due on Monday.

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