微積分公式
4.6 有理函数的积分 习题4.6
求下列不定积分：
x3dx （1）?x?3解：
x327?t39t2?2t?9t?27??dt???27t?27lnt?C?x?3dxx?3?t??t?32??9?x?3??27?x?3??27lnx?3?C.322x?3（2）?2dx
x?3x?102x?31解：?2dx??2d?x2?3x?10??lnx2?3x?10?C.
x?3x?10x?3x?10x?1（3）?2dx
x?2x?5?解：
2d?x?1?x?112x?2?41d?x?2x?5??x2?2x?5dx?2?x2?2x?5dx?2?x2?2x?5?2??x?1?2?22?x?3?32
1x?1?ln?x2?2x?5??arctan?C.22（4）
dx?x?x2?1?
2d?x?dx11?11?1x22??22???2?2?C. 解：??d?x??ln22x?1?2x?1x?x?1?2x?x?1?2?x（5）解：
3?x3?1dx
32?x?12x?1?3?1dx??dx?lnx?1?dx?22?x3?1???x?1x?x?12x?x?1??21d?x?x?1?31?lnx?1???dx222x2?x?12??1??3???x????2??2??12x?1?lnx?1?ln?x2?x?1??3arctan?C.23
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微積分公式
（6）
??x?1??x?1?dx
2x2?11?1??2x?11?1122dx???dx?lnx?1??C. 解：??22??x?1x?12x?1?x?1??x?1??x?1?????2（7）解：
xdx??x?1??x?2??x?3?
3??1???2xdx22?dx?????x?1??x?2??x?3???x?1x?2x?3??? ??13?2lnx?2?lnx?1?lnx?3?C.22x5?x4?8dx （8）?3x?x解：
?2x5?x4?8x2?x?8??x3?xdx????x?x?1?x?x?1??x?1???dx??x3x243??8???x??????dx32?xx?1x?1?x3x2???x?8lnx?4lnx?1?3lnx?1?C.32（9）解：
dx??x2?1??x2?x?
111????x??1dx2?22?dx????x2?1??x2?x???xx?1x2?1??? ??112x?2111?lnx?lnx?1??2dx?lnx?lnx?1?ln?x2?1??arctanx?C.24x?1242（10）解：
2
1?x4?1dx
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微積分公式
1dx1?11?dx????dx22?x4?1??x2?1??x2?1?2??x?1x?1??1x?11?ln?arctanx?C.4x?12（11）解：
dx??x2?1??x2?x?1?
dxx?1???x???dx22??x2?1??x2?x?1???x?1x?x?1??112x?11dx ??ln?x2?1???2dx??2222x?x?12?1??3?x??????2???2?1112x?1??ln?x2?1??ln?x2?x?1??arctan?C.2233（12）
??x?1?22?x22?1?22dx
x2?2x?1d?x2?1?dx1dx??2???arctanx??C. 222x?1x?1?x?1?解：
??x?1??x?1??dx???x2?1?2（13）解：
?x2?2?x2?x?1?2dx
??x2?2?x2?x?1?2dx???x2?x?1?x?1?x2?x?1?2dx???dx12x?1?3?dx22?2x?x?12?x?x?1?
21d?x?x?1?3dx?????2?x2?x?122?x2?x?1222??1?3?????x??????2??2??22x?1113dx??arctan??2x2?x?12??x2?x?1?233dx而
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微積分公式
dxx2x2?x?x2?x?1?x2?x?1??x2?x?12dx??x2x2?2x?2?x?2?2??dx22x?x?1?x?x?1???xdx12x?1?3?2?dxx2?x?1?x2?x?12??x2?x?1?2xdx1131?2??dxx2?x?1?x2?x?12x2?x?12??x2?x?1?2
所以
??dx?x2?x?1?4332?2dx2x?1?3?x2?x?13?x2?x?1?arctan2x?12x?1??C.23?x?x?1?3
于是
??x2?2?x2?x?1?2dx
??42x?1x?1arctan?2?C.33?x?x?1?(14)
dx?3?sin2x
dxsec2xdxdtanx1?2tanx?解：????arctan?C. ??2222??3?sinx3secx?tanx3?4tanx233??（15）
dx?3?cosx
1?t22x,dx?dt， 解：令tan?t，则cosx?221?t1?t2x??tan?dxdt1t12??C. ??arctan?C?arctan???3?cosx?2?t22222????dx（16）?
2?sinxx2t2,dx?dt， 解：令tan?t，则sinx?21?t21?t2页脚内容
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微積分公式
x??2tan?1??dxdt22t?122??C. ?2?sinx??1?t?t2?3arctan3?C?3arctan?3????dx（17）?
1?sinx?cosx2t1?t22x,cosx?,dx?dt， 解：令tan?t，则sinx?2221?t1?t1?t2dxdtx??ln1?t?C?ln1?tan?C. ?1?sinx?cosx?1?t2（18）
dx?2sinx?cosx?5
2t1?t22x,cosx?,dx?dt， 解：令tan?t，则sinx?1?t21?t21?t22x??3tan?1??dxdt13t?112?2sinx?cosx?5??3t2?2t?2?5arctan5?C?5arctan?5??C.
????（19）解：
dx?1?3x?1
dx1?3t2?3t?2??dt??6t?3lnt?C?1?3x?11?x?1?t3??t2???31?x?12?3?2
?61?3x?1?3ln1?3x?1?C.??（20）解：
?3???1dx
x3x?12??32dxx?1?t2t?4t?6t?5??dt?x?1??t??t48t3???6t2?10t?4lnt?C23?x??1
??x?12?4?8?x?13?3?6?x?1?10?2?x?1?4ln??x?1?C.?（21）
?x?1?1dx
x?1?1页脚内容
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微积分刘迎东编第四章习题4.6答案
