忘忧草悠悠
幼苗
共回答了23个问题采纳率:91.3% 举报
都用分部积分
第一题:
∫ln(1+x^2)dx = ln(1+x^2)x -∫xd(ln(1+x^2))
= ln(1+x^2)x - ∫2x^2/(1+x^2)dx
= ln(1+x^2)x -∫[2 -2/(1+x^2)]dx
= ln(1+x^2)x - 2x + 2arctanx + C
第二题:
∫xlnxdx = ∫lnxd(x^2/2)
= 1/2x^2lnx - 1/2∫x^2d(lnx)
= 1/2x^2lnx - 1/2∫xdx
=1/2x^2lnx - 1/4x^2 +C
第三题:
∫x^2(lnx)^2dx = 1/3∫(lnx)^2dx^3
= 1/3x^3(lnx)^2 - 1/3∫x^3d[(lnx)^2]
= 1/3x^3(lnx)^2 - 2/3∫x^2lnxdx
= 1/3x^3(lnx)^2 - 2/9∫lnxdx^3
= 1/3x^3(lnx)^2 - 2/9[x^3lnx - ∫x^2dx]
= 1/3x^3(lnx)^2 - 2/9x^3lnx - 2/27x^3 + C
第四题:
∫lnx/x^2dx = ∫lnx d(-1/x)
= -lnx/x + ∫1/x^2 dx
= -lnx/x - 1/x + C
1年前
10