### 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.) .