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∫(x^2)/(1+x^2)^2dx
题目详情
∫(x^2)/(1+x^2)^2dx
▼优质解答
答案和解析
∫ 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
= ∫ 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
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