求不定积分1题,1/[x+√(1-x∧2)]

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/26 00:30:06
求不定积分1题,1/[x+√(1-x∧2)]求不定积分1题,1/[x+√(1-x∧2)]求不定积分1题,1/[x+√(1-x∧2)]代换x=sintπ/2>t>-π/2cost>0dx=costdt∫

求不定积分1题,1/[x+√(1-x∧2)]
求不定积分1题,
1/[x+√(1-x∧2)]

求不定积分1题,1/[x+√(1-x∧2)]
代换x=sint π/2>t>-π/2 cost>0
dx=costdt
∫1/[x+√(1-x^2)]dx
=∫cost/(sint+cost)dt
I=∫cost/(sint+cost)dt
J=∫sint/(sint+cost)dt
I+J=∫(sint+cost)/(sint+cost)dt=∫dt=t=arcsinx+C
J-I=∫(cost-sint)/(sint+cost)dt=
=∫d(sint+cost)/(sint+cost)dt
=ln(sint+cost)
=ln(x+√(1-x^2))+C
I=1/2*((I+J)-(J-I))
=1/2arcsinx-1/2ln(x+√(1-x^2))+C