Part III: Trigonometric Substitution Trigonometric Substitution Rules 1) For ?(a
ID: 3288559 • Letter: P
Question
Part III: Trigonometric Substitution
Trigonometric Substitution Rules
1)For ?(a^2-u^2 ) let u = asin?? and ?(a^2-u^2 ) = acos??
2)For ?(a^2+ u^2 ) let u = atan? and ?(a^2+u^2 ) = asec?
3)For ?(u^2-a^2 ) let u = asec?? and ?(u^2-a^2 ) = atan?
1. Integrate??dx/?(25+4x^2 )Identify the following:
a = u = du =
What adjustment, if any, needs to be made to the integral to complete the substitution in to u?
2. Rewrite the above integral so that it is all in u.
3. Which of the trig substitution rules above will we use to solve this integral? Why? Draw and label the relevant right triangle diagram. See the week 2 lecture or Section 28.8, page 858-9 for ideas on how to do this
4. Use the rule you chose and the diagram to identify what you will replace u, du and?(a^2+u^2 )with.
5. Rewrite the integral from step 2 using the replacements from step 4. Did you use all the replacements? If not, why not? Can this integral be simplified? If so, do it?
Explanation / Answer
Note that
a cos ? + b sin ? = (a, b)