23.(本小题6分)
设z=z(x，y)是由方程2sin(2x+3y-5z)=2x+3y-5z所确定的隐函数，求证
证：令F(x,y,z)?2x?3y?5z?2sin(2x?3y?5z),则
?z?z??1. ?x?yFx?(x,y,z)??2x?x???3y?x???5z?x???2sin(2x?3y?5z)?x??2?4cos?2x?3y?5z?
Fy?(x,y,z)??2x?y???3y?y???5z?y???2sin(2x?3y?5z)?y??3?6cos?2x?3y?5z? Fz?(x,y,z)??2x?z???3y?z???5z?z???2sin(2x?3y?5z)?y???5?10cos?2x?3y?5z?
从而
Fy?2?4cos?2x?3y?5z?3?6cos?2x?3y?5z?F??z?z，， ??x???????xFz??5?10cos?2x?3y?5z??yFz??5?10cos?2x?3y?5z?2?4cos?2x?3y?5z?3?6cos?2x?3y?5z??5?10cos?2x?3y?5z??z?z??????1，原题得证. ?x?y5?10cos?2x?3y?5z??5?10cos?2x?3y?5z??5?10cos?2x?3y?5z?24.(本小题7分) 计算定积分I=?0解：令t?38811?3xdx.
x，则x?t3；当x?0时，t?0;当x?8时，t?2;
12222t2t?1?12?t?1??t?1??12?131?从而，I=?0dx=dt?3dt?3dt?3dt?3t?1?dt ??3???????000001?x1?t1?t1?t1?t1?t????1?1???1???3?t2?t?lnt?1??3????22?2?ln?2?1?????02?0?ln?0?1????3ln3
?2?0??2????22