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Part III: Trigonometric Substitution Trigonometric Substitution...

Part III: Trigonometric Substitution

Trigonometric Substitution Rules

For √(a^2-u^2 ) let u = asinθ and √(a^2-u^2 ) = acosθ

For √(a^2+ u^2 ) let u = atanθ and √(a^2+u^2 ) = asecθ

For √(u^2-a^2 ) let u = asecθ and √(u^2-a^2 ) = atanθ

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?

6. Integrate the resulting trig function using a basic formula.

7. Use the information from steps 1 and 4 to rewrite the answer in step 6 in terms of the variable x.

u = 2x = 5tanθ and √(a^2+u^2 )=√(25+4x^2 )=5secθ

8. How does knowing the trigonometric relationships as defined by the three sides of a right triangle and the Pythagorean Theorem help us solve integrals using trig substitutions?

Now try these

9. ∫▒dx/(x√(49-9x^2 )) 10. ∫▒dx/√(16x^2-1)

Trigonometric Substitution Rules

For √(a^2-u^2 ) let u = asinθ and √(a^2-u^2 ) = acosθ

For √(a^2+ u^2 ) let u = atanθ and √(a^2+u^2 ) = asecθ

For √(u^2-a^2 ) let u = asecθ and √(u^2-a^2 ) = atanθ

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?

6. Integrate the resulting trig function using a basic formula.

7. Use the information from steps 1 and 4 to rewrite the answer in step 6 in terms of the variable x.

u = 2x = 5tanθ and √(a^2+u^2 )=√(25+4x^2 )=5secθ

8. How does knowing the trigonometric relationships as defined by the three sides of a right triangle and the Pythagorean Theorem help us solve integrals using trig substitutions?

Now try these

9. ∫▒dx/(x√(49-9x^2 )) 10. ∫▒dx/√(16x^2-1)

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