1?6t21?原式=?dx??dt?61?dt………………………..…………….1 2??32001?t20??1?t?1?x3?=6? ………………………..………………….1
2121?sinx??cosx?x-sinx5.解：由题意得. f(x)=?........................................ 1 ?=2xx???xf?(x)dx=?xdf(x)=xf(x)-?f(x)dx ........................................ 2
cosx?x-sinxsinx-+C ........................................ 1
x2x2sinx+C. ....................................... 1 =cosx-x=x?1dy(arctant)?1?t21???? ........................................ 2 6. 解：
tdx[ln1?t2]?t21?t1)??1(22dy1?ttt????3? ..............................................................3
tdx2[ln1?t2]?t1?t22四、证明：
?a?af(x)dx??f(x)dx??f(x)dx ..................................................... 1
?a00aaa000a令x??t,?f(x)dx??f(?t)d(?t)??f(?t)dt??f(?x)dx ..................2
?a0??f(x)dx=?f(x)?f(?x)dx ..................................................... 1
-a0aa?cosxcosxcosx4dx?(???41?ex?01?ex?1?e-x)dx ..................................................... 2
4?? ? =?40cosxdx ..................................................... 1
2 ..................................................... 1 2五、解：令x?tx0?u
xxx?tf?x?t?dt??0?x?u?f?u?du?x?0f?u?du??0uf?u?du.............................2
所以
?0f?t?dt?x?x?0f?u?du??0uf?u?du
xxx求导得：
f?x??1??0f?u?du?xf?x??xf?x??1??0f?u?du .........................2
xx第 6 页 共 7 页
求导:
f??x??f?x? ............................ ................................. ......................................1
?f?x??cex............................ ................................. .....................................1
由等式f?x??1??0f?u?du得f?0??1 则c?1 ?fx?x??xe................2
六、解：（1）A=?10y2dy................... ................................. ......................................2 2y311=................... ................................. ......................................1
606（2）V=
?120?(1-2x)2dx ................. ......................................2
1423?x2+x2)2=................. ......................................2 =?(x-3120七、证明：
由积分中值定理：????0,?，使得
??1?3?f(1)?3?e1231-x0221f(x)dx?3?e1-?f(?)(?0)?e1-?f(?) ......................................... 2
3设 F(x)?e1?xf(x) .................................................... 2 则F(x)在??,1?上连续，在??,1?内可导
且F(?)?f(1)?F(1) .................................................... 2
由罗尔定理，至少存在一点???0,??(0,1)使 F?(?)?0 .................................. 2 即f'(?)?2?f(?) .................................................... 1
2??1?3?第 7 页 共 7 页