Integration Using Trigonometric Substitution Brought to you by Tutorial Services – The Math...
-
Upload
franklin-orourke -
Category
Documents
-
view
221 -
download
0
Transcript of Integration Using Trigonometric Substitution Brought to you by Tutorial Services – The Math...
Integration Using Integration Using Trigonometric Trigonometric SubstitutionSubstitutionBrought to you by
Tutorial Services – The Math Center
To eliminate radicals in the integrand using Trigonometric Substitution
For integrals involving use u = a sin
For integrals involving use u = a tan
For integrals involving use u = a sec
Objective
22 ua
22 au
22 ua
For integrals involving 22 ua
Let u = a sin
Inside the radical you will have
Using the Pythagorean Identities, that is equal to
This will result in = a cos22 ua
)sin²-a²(1
)a²(cos²
For integrals involving
Let u = a tan
Inside the radical you will have
Using the Pythagorean Identities, that is equal to
This will result in = a sec
22 ua
22 ua
)tan²-a²(1
)a²(sec²
For integrals involving
Let u = a sec
Inside the radical you will have
Using the Pythagorean Identities, that is equal to
This will result in = + a tan
Positive if u > a, Negative if u < - a
22 au
22 au
)sec²-a²(1
)a²(tan²
Converting LimitsConverting Limits
By converting limits, you avoid By converting limits, you avoid changing back to x, after you are changing back to x, after you are done with the integrationdone with the integration
Because has the formBecause has the form
then then u = xu = x, , a = 3a = 3, and , and x = 3 sinx = 3 sin
29 x 22 ua
Converting LimitsConverting Limits
Now, when x = 0, the Lower Limits is:Now, when x = 0, the Lower Limits is:
0 = 3 sin 0 = 3 sin
0 = sin 0 = sin
0 = 0 =
Now, when x = 3, the Upper Limit is:Now, when x = 3, the Upper Limit is:
3 = 3 sin 3 = 3 sin
1 = sin 1 = sin
/2 = /2 =
ExamplesExamples
►Solve the following integrals: Solve the following integrals:
22 9 xx
dx
14 2x
dx
xdxx 32
Integration Using Trigonometric Integration Using Trigonometric Substitution LinksSubstitution Links
► Integration Using Trigonometric Integration Using Trigonometric Substitution HandoutSubstitution Handout
►Trigonometric Identities HandoutTrigonometric Identities Handout► Integrals and Derivatives HandoutIntegrals and Derivatives Handout►Trigonometric Substitution QuizTrigonometric Substitution Quiz