### SOLUTIONS TO INTEGRATION BY PARTS

SOLUTION 16 : Integrate . Note that

.

Now let

and

so that

and .

Therefore,

.

SOLUTION 17 : Integrate . Note that

.

Let

and

so that

and .

Therefore,

.

SOLUTION 18 : Integrate . Note that

.

Let

and

so that

and .

Therefore,

+ C

+ C .

SOLUTION 19 : Integrate . Note that

.

Let

and

so that

and .

Therefore,

.

SOLUTION 20 : Integrate . Note that

.

Let

and

so that

and .

Therefore,

.

SOLUTION 21 : Integrate . Note that

.

Let

and

so that

and .

Therefore,

.

SOLUTION 22 : Integrate . 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 23 : 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.)

.