(1+x的平方)的平方分之x的平方的不定积分是多少
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(1+x的平方)的平方分之x的平方的不定积分是多少
![(1+x的平方)的平方分之x的平方的不定积分是多少](/uploads/image/z/19846887-15-7.jpg?t=%EF%BC%881%2Bx%E7%9A%84%E5%B9%B3%E6%96%B9%EF%BC%89%E7%9A%84%E5%B9%B3%E6%96%B9%E5%88%86%E4%B9%8Bx%E7%9A%84%E5%B9%B3%E6%96%B9%E7%9A%84%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E6%98%AF%E5%A4%9A%E5%B0%91)
∫ x²/(1 + x²)² dx,令x = tanz,dx = sec²z dz
= ∫ tan²z/sec⁴z · sec²z dz
= ∫ tan²z · cos²z dz
= ∫ sin²z dz
= (1/2)∫ (1 - cos2z) dz
= (1/2)[z - (1/2)sin2z] + C
= (1/2)arctan(x) - (1/2)[x/√(1 + x²)][1/√(1 + x²)] + C
= (1/2)arctan(x) - x/[2(1 + x²)] + C
= ∫ tan²z/sec⁴z · sec²z dz
= ∫ tan²z · cos²z dz
= ∫ sin²z dz
= (1/2)∫ (1 - cos2z) dz
= (1/2)[z - (1/2)sin2z] + C
= (1/2)arctan(x) - (1/2)[x/√(1 + x²)][1/√(1 + x²)] + C
= (1/2)arctan(x) - x/[2(1 + x²)] + C