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