(In the denominator use trig identity A from the beginning of this section.)
(Use antiderivative rule 5 and trig identity F from the beginning of this section.) truein
.
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SOLUTION 11 : Integrate . First square the function, getting truein
(Use trig identity G from the beginning of this section.)
.
Now use u-substitution. Let
so that
,
or
.
Substitute into the original problem, replacing all forms of , getting
(Use antiderivative rule 4 on the first integral. Use antiderivative rule 6 on the second integral.)
(Combine constant with since is an arbitrary constant.)
.
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SOLUTION 12 : Integrate . Use u-substitution. Let
so that
.
Substitute into the original problem, replacing all forms of , getting
.
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SOLUTION 13 : Integrate . First rewrite the function (Recall that .), getting
(Now use trig identity F from the beginning of this section.)
.
On the first integral use u-substitution. Rewrite the second integral and use trig identity F again. Let
so that
.
Substitute into the original problem, replacing all forms of , getting
.
Use u-substitution on the first integral. Use antiderivative rule 7 on the second integral. Let
so that
.
Substitute into the original problem, replacing all forms of , getting
.
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SOLUTION 14 : Integrate . Use u-substitution. Let
so that
,
or
.
Substitute into the original problem, replacing all forms of , getting
.
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SOLUTION 15 : Integrate . Let
so that
.
Substitute into the original problem, replacing all forms of , getting
.
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SOLUTION 16 : Integrate . (Hello ! The term is NOT the product of and . It is the functional composition of functions and . ) Use u-substitution. Let
so that
,
or
.
Substitute into the original problem, replacing all forms of , getting
.
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SOLUTION 17 : Integrate . Use u-substitution. Let
so that
.
Substitute into the original problem, replacing all forms of , getting
.
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SOLUTION 18 : Integrate . Use u-substitution. Let
so that
,
or
.
Substitute into the original problem, replacing all forms of , getting
.
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