积分递推公式1/(x^n根号(x^2+1))

Question

积分递推公式1/(x^n根号(x^2+1))

Answer 1

F(n) = Int( 1 / ( x^n * sqrt(1+x^2) ) ,x)
=lnt( x / ( x^(1+n) * sqrt(1+x^2) ) ,x)
=Int( 1 / x^(1+n) ,sqrt(1+x^2) )
=sqrt(1+x^2) / x^(1+n) - Int( sqrt(1+x^2) ,1 / x^(1+n) ) ——————分部积分
=sqrt(1+x^2) / x^(1+n) + Int( (1+n) * sqrt(1+x^2) / x^(2+n) ,x)
=sqrt(1+x^2) / x^(1+n) + (1+n) * Int( (1+x^2) / ( x^(2+n) * sqrt(1+x^2)),x)
=sqrt(1+x^2) / x^(1+n) + (1+n) * ( F(n) + F(n+2) )
Int(y,x) 就是 ∫ ydx的意思