### SOLUTIONS TO INTEGRATION BY PARTS

SOLUTION 1 : Integrate . Let

and

so that

and .

Therefore,

.

SOLUTION 2 : Integrate . Let

and

so that

and .

Therefore,

.

SOLUTION 3 : Integrate . Let

and

so that

and .

Therefore,

.

SOLUTION 4 : Integrate . Let

and

so that

and .

Therefore,

.

SOLUTION 5 : Integrate . Let

and

so that

and .

Therefore,

.

SOLUTION 6 : Integrate . Let

and

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

and .

Therefore,

.

Now use u-substitution. Let

so that

,

or

.

Then

+ C

+ C

+ C .

SOLUTION 7 : Integrate . Let

and

so that

and .

Therefore,

.

SOLUTION 8 : Integrate . Let

and

so that

and .

Therefore,

(Add in the numerator. This will replicate the denominator and allow us to split the function into two parts.)

(Please note that there is a TYPO in the next step. The second "-" sign should be a "+" sign. However, subsequent steps are correct.)

(The remaining steps are all correct.)

.

SOLUTION 9 : Integrate . Let

and

so that

and .

Therefore,

.

Integrate by parts again. Let

and

so that

and .

Hence,

.