Question

积分题目运算,在线等!

∫ x²√a²+x²dx
A={1,2,3},B={0,1,2},A乘以B等于多少
两题，忘高手指教，写下过程
√a²+x²就是根号下a²+x²，先令X=asint

Answer 1

分部积分
∫ x²√a²+x²dx
=1/3x(a²+x²)^(3/2)-∫√a²+x²dx

对∫√a²+x²dx，令x=tant，可化为
∫sec³tdt
=∫sectdtant
=secttant-∫tantdsect
=secttant-∫secttan²tdt
=secttant-∫sect(sec²t-1)dt
=secttant-∫sec³tdx+∫sectdt
=secttant-∫sec³tdt+ln|sect+tant|

2∫sec³tdt=secttant+ln|sect+tant|

∫sec³tdx=(secttant+ln|sect+tant|)/2+C

转换回去：
∫√a²+x²dx=(x√a²+x²+ln(x+√a²+x²))/2+C
所以，∫ x²√a²+x²dx=
1/3x(a²+x²)^(3/2)-(x√a²+x²+ln(x+√a²+x²))/2+C

A={1,2,3},B={0,1,2}
AB=0*1+2*1+3*2=8

Answer 2

好烦，AB=8