(1+x的平方)的平方分之x的平方的不定积分是多少
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/09/02 22:03:43
(1+x的平方)的平方分之x的平方的不定积分是多少
∫ 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