∫(x^2)/(1+x^2)^2dx

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∫(x^2)/(1+x^2)^2dx
∫(x^2)/(1+x^2)^2dx

∫(x^2)/(1+x^2)^2dx
∫ x²/(1 + x²)² dx,令x = tanz,dx = sec²z dz
= ∫ tan²z/sec⁴z * (sec²z dz)
= ∫ sin²z/cos²z * cos²z dz
= ∫ (1 - cos2z)/2 dz
= z/2 - (1/4)sin2z + C
= (1/2)arctanx - (1/2) * x/√(1 + x²) * 1/√(1 + x²) + C
= (1/2)arctanx - x/[2(1 + x²)] + C

∫[dx/(e^x(1+e^2x)]dx ∫1+2x/x(1+x)*dx∫1+2x/x(1+x) * dx ∫(x-1)^2dx, ∫x^1/2dx 1/(x+x^2)dx ∫dx/x^2(1-x^2) ∫dx/x^2(1+x^2) ∫X^2/1-x^2 dx. ∫X^2/1-x^2 dx. ∫1/x^2+x+1dx ∫1/(x^2+x+1)dx ∫ (x+arctanx)/(1+x^2) dx ∫（x^2+1/x^4)dx ∫2 -1|x²-x|dx ∫x^2/1+X dx. ∫ xe^x/(1+x)^2 dx ∫dx/[(1-x)x^2] ∫ x/(1+X^2)dx=