Question

二重积分及多元复合函数解答具体题目如下

Answer 1

求二重积分[1,5]∫dy[y,5]∫ylnxdx
原式=[1,5]∫ydy[y,5]∫lnxdx=[1,5]∫ydy[xlnx-x]∣[y,5]=[1,5]∫y(5ln5-5-ylny+y)dy
=[1,5]∫[5yln(5/e)-y²lny+y²]dy={[5ln(5/e)]∫ydy-∫y²lnydy+∫y²dy}∣[1,5]
={[5ln(5/e)](y²/2)-(1/3)(y³lny-y³/3)+(1/3)y³]∣[1,5]
={[5ln(5/e)](y²/2)-(1/3)(y³lny)+(4/9)y³]∣[1,5]
=(125/2)ln(5/e)-(125/3)ln5+(500/9)-(5/2)ln(5/e)-(4/9)
=60(ln5-1)-(125/3)ln5+496/9
=(55/3)ln5-(44/9)