Solução_Calculo_Stewart_6e

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F. TX.10 SECTION 7.2 TRIGONOMETRIC INTEGRALS ¤ 303 57. A = =2 π/4 −π/4 π/4 0 =1− 0=1 π/4 (cos 2 x − sin 2 x) dx = −π/4 cos 2xdx π/4 1 cos 2xdx=2 2 sin 2x = sin 2x π/4 0 0 59. It seems from the graph that 2π 0 cos 3 xdx=0, since the area below the x-axis and above the graph looks about equal to the area above the axis and below the graph. By Example 1, the integral is sin x − 1 3 sin3 x 2π 0 =0. Note that due to symmetry, the integral of any odd power of sin x or cos x between limits which differ by 2nπ (n any integer) is 0. 61. Using disks, V = π π π/2 sin2 xdx= π π 1 (1 − cos 2x) dx = π 1 x − 1 sin 2x π = π π − 0 − π +0 = π2 π/2 2 2 4 π/2 2 4 4 63. Using washers, V = π/4 π (1 − sin x) 2 − (1 − cos x) 2 dx 0 = π π/4 0 (1 − 2sinx +sin 2 x) − (1 − 2cosx +cos 2 x) dx = π π/4 (2 cos x − 2sinx +sin 2 x − cos 2 x) dx 0 = π π/4 (2 cos x − 2sinx − cos 2x) dx = π 2sinx +2cosx − 1 sin 2x π/4 0 2 0 = π √ 2+ √ √ 2 − 1 2 − (0 + 2 − 0) = π 2 2 − 5 2 65. s = f(t) = t sin ωu 0 cos2 ωu du. Lety =cosωu ⇒ dy = −ω sin ωu du. Then s = − 1 cos ωt y 2 dy = − 1 1 y3 cos ωt = 1 (1 − ω 1 ω 3 3ω cos3 ωt). 67. Just note that the integrand is odd [f(−x) =−f(x)]. 69. Or: If m 6= n, calculate π −π 1 π 1 sin mx cos nx dx = [sin(m − n)x +sin(m + n)x] dx = 1 cos(m − n)x − − 2 −π 2 m − n If m = n,thenthefirst term in each set of brackets is zero. π cos mx cos nx dx = π 1 [cos(m − n)x +cos(m + n)x] dx. −π −π 2 If m 6= n, this is equal to 1 2 sin(m − n)x m − n + π sin(m + n)x =0. m + n −π If m = n,weget π 1 [1 + cos(m + n)x] dx = π 1 x π sin(m + n)x + −π 2 2 = π +0=π. −π 2(m + n) −π π cos(m + n)x =0 m + n −π
F. 304 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION TX.10 7.3 Trigonometric Substitution 1. Let x =3secθ,where0 ≤ θ< π 2 or π ≤ θ< 3π 2 .Then dx =3secθ tan θdθand √ x2 − 9= √ 9sec 2 θ − 9= 9(sec 2 θ − 1) = √ 9tan 2 θ =3|tan θ| =3tanθ for the relevant values of θ. 1 x √ 2 x 2 − 9 dx = 1 9sec 2 θ · 3tanθ 3secθ tan θdθ= 1 9 cos θdθ= 1 sin θ + C = 1 √ x2 − 9 + C 9 9 x Note that − sec(θ + π) =secθ,sothefigure is sufficient for the case π ≤ θ< 3π 2 . 3. Let x =3tanθ,where− π 2
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