Question

x^2乘以根号下1+x^2的不定积分 用分部积分法 怎么做

Answer 1

x=tant,t=arctanx
dx=(sect)^2
dt
S
X^2*根号（x^2+1）dx
=S
(tant)^2*sect
*(sect)^2
dt
=S[(sect)^2-1]*(sect)^3
dt
=S(sect)^5
*dt-S(sect)^3*dt
首先求∫sec^3(x)
dx：记I=∫sec^3(x)
dx，则I
=∫sec(x)*sec^2(x)
dx
=∫sec(x)*[tan(x)]'
dx
=sec(x)*tan(x)-∫[sec(x)]'*tan(x)
dx
=sec(x)*tan(x)-∫[sec(x)*tan(x)]*tan(x)
dx
=sec(x)*tan(x)-∫sec(x)*tan^2(x)
dx
=sec(x)*tan(x)-∫sec(x)*[sec^2(x)-1]
dx
=sec(x)*tan(x)-∫sec^3(x)
dx+∫sec(x)
dx
=sec(x)*tan(x)-I+ln|sec(x)+tan(x)|+C，
所以2I=sec(x)*tan(x)+ln|sec(x)+tan(x)|+C，
I=sec(x)*tan(x)/2+ln|sec(x)+tan(x)|/2+C，C为任意常数
然后求∫sec^5(x)
dx：记J=∫sec^5(x)
dx，则J
=∫sec^3(x)*sec^2(x)
dx
=∫sec^3(x)*[tan(x)]'
dx
=sec^3(x)*tan(x)-∫[sec^3(x)]'*tan(x)
dx
=sec^3(x)*tan(x)-∫3sec^2(x)*[sec(x)*tan(x)]*tan(x)
dx
=sec^3(x)*tan(x)-3∫sec^3(x)*tan^2(x)
dx
=sec^3(x)*tan(x)-3∫sec^3(x)*[sec^2(x)-1]
dx
=sec^3(x)*tan(x)-3∫sec^5(x)
dx+3∫sec^3(x)
dx
=sec^3(x)*tan(x)-3J+3I，
所以4J=sec^3(x)*tan(x)+3I，
J=sec^3(x)*tan(x)/4+3I/4
=sec^3(x)*tan(x)/4+3sec(x)*tan(x)/8+3ln|sec(x)+tan(x)|/8+C，
C为任意常数
再把t=arctanx代入即可