so that

.

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

.

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* SOLUTION 20 :* Integrate
. Use integration by parts. Let

and

so that

and .

Therefore,

.

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* SOLUTION 21 :* Integrate
. Use integration by parts. Let

and

so that

and .

Therefore,

.

Use integration by parts again. Let

and

so that

and .

Hence,

.

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* 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,

.

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* 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.)

.

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* 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.)

.

Click HERE to return to the list of problems.

* SOLUTION 25 :* Integrate
. Use u-substitution. Let

so that

,

or

.

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

.

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* SOLUTION 26 :* Integrate
. Use u-substitution. Let

so that

,

or

.

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

.

Click HERE to return to the list of problems.

* SOLUTION 27 :* Integrate
. First multiply by
, getting

.

.

.

Now use u-substitution. Let

so that

.

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

.

Click HERE to return to the list of problems.

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