### 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.)

.

SOLUTION 2 : Integrate . Use u-substitution. Let

so that

,

or

.

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

(Use antiderivative rule 7 from the beginning of this section.)

.

SOLUTION 3 : Integrate . Use u-substitution. Let

so that

,

or

.

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

(Use antiderivative rule 5 from the beginning of this section.)

.

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

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

.

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 beginning of this section, getting

.

Now use u-substitution. Let

so that

,

or

.

Substitute into the original problem, getting

(Use antiderivative rule 1 from the beginning of this section.)

(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 and use trig identity F from the beginning of this section on the second integral.)

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

.

SOLUTION 7 : Integrate . First rewrite the function (Recall that .), 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.)

.

Now use u-substitution. Let

so that

,

or

.

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

.

SOLUTION 8 : Integrate . Use u-substitution. Let

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

,

or

.

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

.

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

(Now factor the numerator. Recall that .)

(Divide out the factors of .)

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

.