1910-Prelim-3soln-2013 - Mathematics 1910 Prelim 3 December...

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Mathematics 1910, Prelim 3December 5, 2013, 7:30 PM to 9:00 PMYou are NOT allowed calculators, the text or any other book or notes. SHOW ALL WORK!Writing clearly and legibly will improve your chances of receiving the maximum credit that yoursolution deserves. Please label the questions in your answer booklet clearly.Write your name and section numberon each booklet you use and sign the integrity statement onthe cover of the booklet. You may leave when you have finished, but if you have not handed in yourexam booklet and left the room by 8:45pm, please remain in your seat so as not to disturb otherswho are still working. There are5 questions.To receive full credit, you are expected toevaluate integrals, but it is not necessary to simplify the resulting expressions asmuch as possible. Using geometric reasoning to evaluate integrals is also allowed,but explain the reasoning.1. (30pts) Evaluate the following integrals(a)integraldisplayx2sin(x)dxIntegrate by parts (u=x2, v= sin(x)) to obtainintegraldisplayx2sin(x)dx=-x2cos(x) +2integraldisplayxcos(x)dx+C, and then integrate by parts a second time onintegraldisplayxcos(x)dx(u=x, v= cos(x)) to getintegraldisplayx2sin(x)dx=-x2cos(x) + 2xsin(x) + 2 cos(x) +C.(b)integraldisplayπ0sin3(x)dx

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