3. R π/2 0 1 . 120 7. R π/2 11. R π/2 0 0 π . 16 sin7 θ cos5 θdθ = cos2 θdθ = R π/2 0 R π/2 0 1 2 sin7 θ(1−sin2 θ)2 d sin θ = (cos(2θ) + 1) dθ = R π/2 sin2 x cos2 xdx = 0 t−t cos(2t) dt 2 1 4 1 4 sin2 (2x)dx = t2 4 h t sin(2t) 4 R π/2 0 23. R R tan2 xdx = (sec2 x − 1)dx = tan x − x + C. 27. R π/3 0 117 . 8 5 4 tan x sec xdx = R π/3 0 5 − R u7 (1−u2 )2 du = θ π/2 2 0 2 tan x(1+tan x)d tan x = = t2 4 R √3 0 h u8 8 − 2 u10 + − sin(4x) 32 10 u12 12 i1 = π4 . (1 − cos(4x)) dx = i t sin2 tdt = − 1 8 sin(2t) dt 4 R = 0 sin(2θ) + 13. R R1 − t sin(2t) 4 5 h x 8 − cos(2t) 8 2 u (1+u )du = h u6 6 iπ/2 = 0 + C. + u8 8 i √3 = 0 R sin(8x) cos(5x)dx = 12 sin(13x) + sin(3x)dx = − cos(13x) + − cos(3x) + C. 26 6 R 2 2x 56. (a) u(−du) = −u2 = − cos + C. 2 R 2 2 (b) udu = u2 = sin2 x + C 0 . C 0 = C − 12 . R (c) 21 sin(2x)dx = − cos(2x) + C 00 . C 00 = C − 14 . 4 R R R 2 (d) sin x cos xdx = sin x sin x− cos x sin xdx ⇒ sin x cos xdx = sin2 x +C 000 . C 000 = C 0. π/4 R π/4 R π/4 sin(2x) 2 2 57. −π/4 (cos x − sin x)dx = −π/4 cos(2x)dx = 2 = 1. 41. R −π/4 61. Rπ π/2 2 π(sin x) dx = π 2 Rπ π/2 (1 − cos(2x))dx = π 2 h x− sin(2x) 2 iπ π/2 = π2 . 4 h iπ Rπ 68. If m = n, 21 −π (1 − cos(2mx))dx = 12 x − sin(2mx) = π. If m 6= n, 2m h iπ −π n)x − cos(m + n)x)dx = 21 sin(m−n)x − sin(m+n)x = 0. m−n m+n 1 2 Rπ −π (cos(m − −π 70. By 68., 1 π Rπ −π N P an sin(nx) sin(mx)dx = n=1 1 1 π Rπ −π am sin(mx) sin(mx)dx = am . 0 =