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WUSTL MATH 132 - 132_07_f14

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Calculus II PLTLFall 2014Worksheet 7These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Round Robin) Use a substitution to change the integralR[cos−1√x]/√x dx into oneyou can find in the integral table in your text. Then evaluate it.2) (Scribe) EvaluateRπ/202 si nh(sin θ) cos θ dθ.3) (Pairs) Evaluate the integralR1306/√1 + 9x2dx three different ways:(a) Make the substitution u = 3x and then use an integral table to express theintegral in terms of a hyberbolic trigono metric function.(b) Make the sub stitution u = 3x and then u se the correspondence between hy-perboli c trig fun ction s and natural logarithms to express the integral in terms of naturallogs.(c) Make the trigo nometric substitution tan θ = 3x and solve.4) (Scribe) Find a par ametrization defining the movement of a particle moving asdescri bed below.(a) The particle starts at the point (−1, 3) and moves in a straight line to the point(3, −2).(b) The particle starts at (1, 2) and moves counterclockwise around the circle x2+(y − 1 )2= 4.5) (Pairs) Consider the function f(x) = xe−x.(a) Evaluate the definite integralR10f(x) dx.(b) Evaluate the d efinite integralR100f(x) dx.(c) Evaluate the d efinite integralR1000f(x) dx.(d) Evaluate the improper integralR∞0f(x) dx if it is convergent, or explain why itis divergent.6) (Round Robin) Calculate the derivative with respect to x of each of the functionsbelow (x > 0).(a) f(x) = ln(x + 1)(b) f(x) = 2 ln(x + 1)(c) f(x) = (x + 1)−1(d) f(x) = (x + 1)−2(e) f(x) = ln(x + 1) + (x + 1)−1(f) f(x) = ln(x + 1) − (x + 1)−1(g) f(x) = 2 ln(x + 1) + (x + 1)−1(h) f(x) = 2 ln(x + 1) − (x + 1)−1(i) f(x) = ln(x + 1) + (x + 1)−2(j) f(x) = ln(x + 1) − (x + 1)−2How could you use the work you just did to answer the following question?With partial fractions you getR(x + 2)/(x2+ 2 x + 1) dx = f(x) + C. What is


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WUSTL MATH 132 - 132_07_f14

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