Calculus II PLTLFall 2014Worksheet 5These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Round Robin) Find the exact value of the a rea und er the first arch of f(x) = x sin x(graphed below) in the first quadrant.2) (Pairs) Consider the integralRcos2θ dθ.(a) Start off by using integration by parts, with u = cos θ, dv = cos θ dθ, then sol ve.(b) Use a trig identity to rewrite the integrand i n terms of cos(2θ), and then solve.(c) Check that your answers are the same.3) (Round Robin) Find the average value of the function f(x) = 4(x −2)/(x2−4x + 8)over the interval [2, 4].Hint: Rewrite x2− 4x + 8 = (x − a)2+ b, simplify 4(x − 2)/(x2− 4x + 8) so tha t thedenominator of the integrand is of the form u2+ 1, where u i s a function of x, and use thetrigonometric substitution u = tan θ.4) (Pairs) (a) EvaluateRsin2x cos3x dx.(b) EvaluateRsin(2x) cos(3x) dx.5) ( ) Evaluate each integral.(a)Rt sin(5 − t2) dt(b)R2π0p1 −sin2θ dθ. Don’t forget:√x2= |x|.(c)R10u ar csin u2du6) (Round Robin) For each of the following integrals, indicate wh ether integration byparts or integration by substitution is more appropri ate. Explain what you would set equalto u for each integral.(a)Rx/(1 + x2) dx(b)Rx cos x dx(c)Rln x dx(d)Rx2sin(x3) dx(e)R1/√3x + 1
View Full Document