please show work for all the problems 5,9,13,17,27,31,39,41,51,55 nvolving sine
ID: 2856828 • Letter: P
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please show work for all the problems
5,9,13,17,27,31,39,41,51,55
nvolving sine and In defini Finding an ntegra Cosine In Exercises 1-12, find the indefinite integral 1. cos5 x sin x dx cos 3 x sin4 x dx sin 2x cos 2x dx sin 3x dx sin x cos 2 x dx COS cos t sin 20 cos 20 do dt Vsint cos2 3x dx 10 sin4 60 do x2 sin2 x dx 11 x sin2 x dx 12 Using Wallis's Formulas In Exercises 13-18, use Wallis's Formulas to evaluate the integral. /2 /2 E 13 14 cos x cos x dx /2 /2 15. cos 10x dx 16 sin x dx /2 /2 17 sin x dx 18 sin x dx Finding an Indefinite Integral Involving Secant and Tangent In Exercises 19-32, find the indefinite integral. 20 sec4 2x dx 19 sec 4x dx tan 3x dx 21 sec Tx dx 22 23 24 tan tan Sec 25 tan 2t sec3 2t dt 26 tans 2x sec4 2x d.x 27 sec6 4x tan 4x dx 28 Sect tan dx 29 sec5 x tan x dx 30 tan 3x dx tan x tan X 31 32 Sec X Sec X Differential Equation In Exercises 33-36, solve the differential equation 35. y' tan 3x sec 3x Slope Field In Exercises 37 and 38, a differential equation a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to MathGraphs.com sec2 x tan2 x, 00,-) dy dy sin2x, (0,0) 37 38 1.5 1.5 1.5 1.5 Slope Field In Exercises 39 and 40, use a computer algebra system to graph the slope field for the differential equation, and graph the solution through the specified initial condition. dy 3 sin x dy 3 y tan2 x, y(0) 3 y(0) 2 39, 40 Using Product-to-Sum Identities In Exercises 41-46 find the indefinite integral. 41 cos 2x cos 6x dx 42 cos 50 cos 30 do 44 (-7x) cos 6x dx Sin 43 sin 2x cos 4x dx 45, sin 6 sin 30 d6 46 sin 5x sin 4x dx Finding an Indefinite Integral In Exercises 47-56, find the indefinite integral. Use a computer algebra system to confirm your result 47 cot 2x dx 48 Sec 49, csc4 3x dx 50 cott csc cot t cot t 51 dt 52 dt csc t csc t Sin X cos x 53 54 sec x tan x cos xExplanation / Answer
5) sin3x cos2x dx
==> sinx sin2x cos2x dx
==> sinx (1 - cos2x) cos2x dx since sin2x + cos2x = 1
==> sinx (cos2x - cos4x) dx
substitute cosx = u ==> -sinx dx = du
==> sinx (cos2x - cos4x) dx = -(u2 - u4) du
==> u4 - u2 du
==> u4+1/(4 +1) - u2+1/(2 +1) + c since xn dx = xn+1/(n +1)
==> u5/5 - u3/3 + c
substitute back u = cosx
==> cos5x/5 - cos3x/3 + c
Hence sin3x cos2x dx = cos5x/5 - cos3x/3 + c
9) cos2 (3x) dx
==> [1 + cos (2*3x)]/2 dx since cos2a = (1 + cos(2a))/2
==> (1/2) [1 + cos(6x)] dx
==> (1/2) [ dx + cos(6x) dx]
==> (1/2) [x + (1/6) sin(6x) ] + c since cos(ax) dx = (1/a) sin(ax) + c
==> x/2 + (x/12) sin(6x) + c
==> (1/12)[6x + sin(6x)] + c
Hence cos2 (3x) dx = (1/12)[6x + sin(6x)] + c
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