### 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,     .