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6 Nov 2019
Trigonometric Substitution Rules
For v(a^2-u^2 ) let u = asin?? and v(a^2-u^2 ) = acos??
For v(a^2+ u^2 ) let u = atan? and v(a^2+u^2 ) = asec?
For v(u^2-a^2 ) let u = asec?? and v(u^2-a^2 ) = atan?
1. Integrate ?dx/v(25+4x^2 ). Identify the following:
a = u = du =
What adjustment, if any, needs to be made to the integral tocomplete 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 solvethis integral? Why? Draw and label the relevant right trianglediagram.
4. Use the rule you chose and the diagram to identify what you willreplace u, du and v(a^2+u^2 ) with.
5. Rewrite the integral from step 2 using the replacements fromstep 4. Did you use all the replacements? If not, why not? Can thisintegral be simplified? If so, do it?
6. Integrate the resulting trig function using a basicformula.
7. Use the information from steps 1 and 4 to rewrite the answer instep 6 in terms of the variable x.
u = 2x = 5tan? and v(a^2+u^2 )=v(25+4x^2 )=5sec?
8. How does knowing the trigonometric relationships as defined bythe three sides of a right triangle and the Pythagorean Theoremhelp us solve integrals using trig substitutions?
9. ?dx/(xv(49-9x^2 )) 10. ?dx/v(16x^2-1)
Trigonometric Substitution Rules
For v(a^2-u^2 ) let u = asin?? and v(a^2-u^2 ) = acos??
For v(a^2+ u^2 ) let u = atan? and v(a^2+u^2 ) = asec?
For v(u^2-a^2 ) let u = asec?? and v(u^2-a^2 ) = atan?
1. Integrate ?dx/v(25+4x^2 ). Identify the following:
a = u = du =
What adjustment, if any, needs to be made to the integral tocomplete 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 solvethis integral? Why? Draw and label the relevant right trianglediagram.
4. Use the rule you chose and the diagram to identify what you willreplace u, du and v(a^2+u^2 ) with.
5. Rewrite the integral from step 2 using the replacements fromstep 4. Did you use all the replacements? If not, why not? Can thisintegral be simplified? If so, do it?
6. Integrate the resulting trig function using a basicformula.
7. Use the information from steps 1 and 4 to rewrite the answer instep 6 in terms of the variable x.
u = 2x = 5tan? and v(a^2+u^2 )=v(25+4x^2 )=5sec?
8. How does knowing the trigonometric relationships as defined bythe three sides of a right triangle and the Pythagorean Theoremhelp us solve integrals using trig substitutions?
9. ?dx/(xv(49-9x^2 )) 10. ?dx/v(16x^2-1)
For v(a^2-u^2 ) let u = asin?? and v(a^2-u^2 ) = acos??
For v(a^2+ u^2 ) let u = atan? and v(a^2+u^2 ) = asec?
For v(u^2-a^2 ) let u = asec?? and v(u^2-a^2 ) = atan?
1. Integrate ?dx/v(25+4x^2 ). Identify the following:
a = u = du =
What adjustment, if any, needs to be made to the integral tocomplete 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 solvethis integral? Why? Draw and label the relevant right trianglediagram.
4. Use the rule you chose and the diagram to identify what you willreplace u, du and v(a^2+u^2 ) with.
5. Rewrite the integral from step 2 using the replacements fromstep 4. Did you use all the replacements? If not, why not? Can thisintegral be simplified? If so, do it?
6. Integrate the resulting trig function using a basicformula.
7. Use the information from steps 1 and 4 to rewrite the answer instep 6 in terms of the variable x.
u = 2x = 5tan? and v(a^2+u^2 )=v(25+4x^2 )=5sec?
8. How does knowing the trigonometric relationships as defined bythe three sides of a right triangle and the Pythagorean Theoremhelp us solve integrals using trig substitutions?
9. ?dx/(xv(49-9x^2 )) 10. ?dx/v(16x^2-1)
Lelia LubowitzLv2
17 Feb 2019
