灵鹊做题网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
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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
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
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/(πR^2)]∫(1-r/R)^(1/n)rdr
= (2u/R^2)∫ t[-nRt^(n-1)]R(1-t^n)dt
= (2nu)∫t^n(1-t^n)dt
= (2nu)∫[t^n-t^(2n)]dt
= (2nu)[t^(n+1)/(n+1)-t(2n+1)/(2n+1)]
= 2nu*n/[(n+1)(2n+1)]
= 2n^2u/[(n+1)(2n=1)].
于是 [2πu/(πR^2)]∫(1-r/R)^(1/n)rdr
= (2u/R^2)∫ t[-nRt^(n-1)]R(1-t^n)dt
= (2nu)∫t^n(1-t^n)dt
= (2nu)∫[t^n-t^(2n)]dt
= (2nu)[t^(n+1)/(n+1)-t(2n+1)/(2n+1)]
= 2nu*n/[(n+1)(2n+1)]
= 2n^2u/[(n+1)(2n=1)].