Calculus II PLTL Fall 2014 Worksheet 5 These problems are to be done without the use of a calculator unless otherwise specified 1 Round Robin Find the exact value of the area under the first arch of f x x sin x graphed below in the first quadrant R 2 Pairs Consider the integral cos2 d a Start off by using integration by parts with u cos dv cos d then solve b Use a trig identity to rewrite the integrand in terms of cos 2 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 that the denominator of the integrand is of the form u2 1 where u is a function of x and use the trigonometric substitution u tan R 4 Pairs a Evaluate sin2 x cos3 x dx R b Evaluate sin 2x cos 3x dx 5 Evaluate each integral R t sin 5 t2 dt R 2 p b 0 1 sin2 d Don t forget x2 x R1 c 0 u arcsin u2 du a 6 Round Robin For each of the following integrals indicate whether integration by parts or integration by substitution is more appropriate Explain what you would set equal to u for each integral R a x 1 x2 dx R b x cos x dx R c ln x dx R 2 d x sin x3 dx R e 1 3x 1 dx R x f xe dx
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