(4)J0 +∞m1dx (5)+m (6)0+0m+; (7)+x 4.讨论下列积分的收敛性与绝对收敛性 (1)Jo sin z-dar (2)0+∞dx; (3)0++dr(q≥0); (4)too sin(+eda 5.计算下列瑕积分的值 (1)Jo(n ) (2)1= 6.证明积分A=J3l(inx)dr收敛,并求其值 7.利用上题结果,证明 (1)Jo 8 In(sin e)de=2-In 2: (2)Jo Iside=2r In2; (3)J sin20In(sin 0)d0=T(2-In 2); (4)+d=3ln2(4) R +∞ 0 ln(1+x) xp dx; (5) R +∞ 1 dx xp lnq x ; (6) R +∞ 0 dx xp+xq ; (7) R +∞ 0 dx √3 x(x−1)2(x−2) ; (8) R 0 −∞ e x ln |x|dx. 4．讨论下列积分的收敛性与绝对收敛性: (1) R +∞ 0 sin x 2dx; (2) R +∞ 0 sinp x xq dx; (3) R +∞ 0 x p sin x 1+xq dx(q ≥ 0); (4) R +∞ 0 sin(x+ 1 x ) xn dx. 5．计算下列瑕积分的值: (1) R 1 0 (ln x) ndx; (2) R 1 0 x n √ 1−x dx. 6．证明积分A = R π 2 0 ln(sin x)dx收敛,并求其值. 7．利用上题结果，证明: (1) R π 0 θ ln(sin θ)dθ = −π 2 2 ln 2; (2) R π 0 θ sin θ 1−cos θ dθ = 2π ln 2; (3) R π 2 0 sin2 θ ln(sin θ)dθ = π 4 ( 1 2 − ln 2); (4) R 1 0 ln(1+x) 1+x2 dx = π 8 ln 2. 5