Question

高数求微积分

紧急求大虾们第二部到第三部怎么计算得到。。。。。求给出稍微详细点的解题步骤

Answer 1

泰勒展开式。

Answer 2

令 (1-r/R)^(1/n) = t, 则 1-r/R = t^n, r=R(1-t^n), dr=-nRt^(n-1)dt
于是 [2πu<max>/(πR^2)]∫<0,R>(1-r/R)^(1/n)rdr
= (2u<max>/R^2)∫<1,0> t[-nRt^(n-1)]R(1-t^n)dt
= (2nu<max>)∫<0,1>t^n(1-t^n)dt
= (2nu<max>)∫<0,1>[t^n-t^(2n)]dt
= (2nu<max>)[t^(n+1)/(n+1)-t(2n+1)/(2n+1)]<0,1>
= 2nu<max>*n/[(n+1)(2n+1)]
= 2n^2u<max>/[(n+1)(2n=1)].