### SOLUTIONS TO U-SUBSTITUTION

SOLUTION 10 : Integrate . First rewrite the function by multiplying by , getting

(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

.

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.)

.

SOLUTION 12 : Integrate . Use u-substitution. Let

so that

.

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

.

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

.

SOLUTION 14 : Integrate . Use u-substitution. Let

so that

,

or

.

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

.

SOLUTION 15 : Integrate . Let

so that

.

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

.

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

.

SOLUTION 17 : Integrate . Use u-substitution. Let

so that

.

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

.

SOLUTION 18 : Integrate . Use u-substitution. Let

so that

,

or

.

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

.