solutions to trigonometric integrals

8/3/2019 Solutions to Trigonometric Integrals

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SOLUTIONS TO TRIGONOMETRIC INTEGRALS

SOLUTION 1 : Integrate . Use u-substitution. Let

so that

,

or

.

Substitute into the original problem, replacing all forms of , getting

(Use antiderivative rule 2 from the beginning of this section.)

.

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so that

,

or

.



.



so that

,


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or

.



.


SOLUTION 4 : Integrate . Begin by squaring the function,getting

(Use trig identity A from the beginning of this section.)


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.

Now use u-substitution. Let

so that.

Substitute into the original problem, replacing all forms of x, getting

.


SOLUTION 5 : Integrate . First use trig identity C from the beginningof this section, getting


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.


so that

,

or

.

Substitute into the original problem, getting



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(Combine constant with since is an arbitrary constant.)

.


SOLUTION 6 : Integrate . Begin by squaring the function, getting

(Use antiderivative rule 7 from the beginning of this section on the first integral anduse trig identity F from the beginning of this section on the second integral.)


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(Now use antiderivative rule 3 from the beginning of this section.)

.


SOLUTION 7 : Integrate . First rewrite the function (Recallthat .), getting

(Now use trig identity A from the beginning of this section.)

(Use antiderivative rule 2 from the beginning of this section on the first integral.)


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.


so that

,

or

.


.




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so that (Don't forget to use the chain rule when differentiating .)

,or

.


.


SOLUTION 9 : Integrate . First use trig identity A from the beginningof this section to rewrite the function, getting


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solutions to trigonometric integrals

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