第十七次课 20环境工程1环境科学1、2班113(124)人 ①第一换元法积分基本原理(凑微分法积分) ②第一换元法积分举例 ------------------------------++-
???f???x?????x?F??u??f?u???F?x???????f?u?du?F?u??C
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1sin?2024x?2024dx?2024?11?sin2024xd2024x??cos?2024x??C????2024?2024
11?cosmxdx?m?cosmxd?mx??msin?mx??C?1sinkxdx?coskx?C?k?sin?2024x?dx??sinxcosxdx??sinx?dsinx??u?du?m(ax?b)dx(m??1).?12sinx?C 211mm(ax?b)adx?(ax?b)d?ax?b???aa1m?11、 ??ax?b??Ca?m?1??2x3x?5?dx???3?3x2?5?244?C1?dxa2?x2?1adxd??x?a??1?a???1arctanx?C1???x?2a??x?2aa?a??1???a???dxa2?x2?1aarctanxa?C?43?2x2dx?423arctan2x3?C?11x2?4x?6dx???dx?1arctanx?2?Cx?2?2?2222?2x?5x2?4x?6dx??2x?41x2?4x?6dx??x2?4x?6dx??ln?x2?4x?6??1x?22arctan2?C?2024x?10020242x2?4x?6dx?x?42?x2?4x?6dx?3940?1x2?4x?6dx2、?20242ln?x2?4x?6??39401x?22arctan2?C?dxa2?x2?arcsinxa?C(a?0).1d??x???adx?a??2(a?0).??(a?0).1???x??x2?a??1????a???arcsinxa?C?16?3x2dx?13arcsin3x6?C?112dx??dx?arcsinx?17?2x?x8??x?1?222?C?2x?32dx????2x?27?2x?x2dx??17?2x?x7?2x?x2dx???27?2x?x2?arcsinx?13、
22?C
4
、
sinx?sinxdcosxdx???dx?????lncosx?Ccosxcosxcosxcosx1cotxdx?lnsinx?C?dx???sinx?sinxdsinx?lnsinx?C
dx1x?a2?x2?aarctana?Cdxx(a?0)?arcsin?C?a2?x2a?tanxdx??lncosx?C??5、cotxdx??
??dx1x?a?ln?Cx2?a22ax?adx1x?a?ln?a2?x22ax?a?C1?11???dx?2a??x?ax?a??1111dx?dx6、?2a?x?a2a?x?a11?lnx?a?lnx?a?C2a2a1x?a?ln?C2ax?a?
?d?x?a?1dx???lnx?a?Cx?ax?a?cscxdx?lncscx?cotx?C?secxdx?lnsecx?tanx?Cdx1x?a?ln?a2?x22ax?a?C?cscxdx?lncscx?cotx?C??1sinxdcosxdx??dx???22?sinxsinx1?cosx21cosx?11?cosx?1???ln?C??ln?C2cosx?12sin2x??ln?cosx?1?sinx?C??lncotx?sinx?C21cosx?11?cosx?1?1?cosx?ln?C?ln?C?ln?C 2cosx?12sin2xsinx?lncscx?cotx?C232222sinxcosxdx?sinxcosxcosxdx?sinxcosx?dsinx???sin3xsin5x??sinx?1?sinx??dsinx???sinx?sinx?dsinx???C35
cos3x3?sinxdx??cosx?3?Csin4x3?sinxcosxdx?4?C2224?1?cos2x?4sinxdx??dx???2??1???1?2cos2x?cos22x?dx41111?cos4x?x?sin2x??dx 44421111?x?sin2x?x?sin4x?C448321?cos2x1112sinxdx?dx?1?cos2xdx?x?sin2x?C????2?2247、
2??dx12dx1d?1?2lnx?1?????ln1?2lnx?C
x(1?2lnx)2x(1?2lnx)2(1?2lnx)2e323e3x22dx??dx??e3xd3x?e332x33xx8、
??x?C
9、secxdx.?6
6424secxdx?secxsecxdx?sec???xdtanx???sec2x?dtanx???1?tan2x?dtanx22???1?2tan2x?tan4x?dtanx2tan5x3?tanx?tanx??C35cot3x?cscxdx???cscxdcotx????1?cotx?dcotx??cotx?3?C422
d?1?e?x?dx?e?xdx10、?????????ln?1?e?x??C x?x?x1?e1?e1?e11、secxdx. 12、
13、cosxdx. 14、sin?ax?b?dx
??4?x12x1dx21dx?dx??arctanx2?C 15、?224??1?x21??x2?21??x2?2?sinaxcosbxdx?作业?sinaxsinbxdx?
?cosaxcosbxdx?16
?1x?1?x?dx
17、
lnlnx?xlnxdx
18、19、
??a2?x2dx(a?0). dxx?adxx?a2222(a?0).
20、
?(a?0).