### SOLUTIONS TO U-SUBSTITUTION

SOLUTION 19 : Integrate . Use u-substitution. Let

so that

.

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

.

SOLUTION 20 : Integrate . Use integration by parts. Let

and

so that

and .

Therefore,

.

SOLUTION 21 : Integrate . Use integration by parts. Let

and

so that

and .

Therefore,

.

Use integration by parts again. Let

and

so that

and .

Hence,

.

SOLUTION 22 : Integrate . Use u-substitution. Let

so that

.

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

.

Now use integration by parts. Let

and

so that

and .

Hence,

.

SOLUTION 23 : Integrate . Use integration by parts. Let

and

so that

and .

Therefore,

.

Use integration by parts again. let

and

so that

and .

Hence,

.

To both sides of this "equation" add , getting

.

Thus,

(Combine constant with since is an arbitrary constant.)

.

SOLUTION 24 : Integrate . Use integration by parts. Let

and

so that

and .

Therefore,

.

Use integration by parts again. let

and

so that

and .

Hence,

.

From both sides of this "equation" subtract , getting

.

Thus,

(Combine constant with since is an arbitrary constant.)

.

SOLUTION 25 : Integrate . Use u-substitution. Let

so that

,

or

.

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

.

SOLUTION 26 : Integrate . Use u-substitution. Let

so that

,

or

.

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

.

SOLUTION 27 : Integrate . First multiply by , getting

.

.

.

Now use u-substitution. Let

so that

.

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

.