同济大学第六版高等数学上下册课后习题
答案4-3
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习题4?3
求下列不定积分: 1. ?xsinxdx; 解
?xsinxdx???xdcosx??xcosx??cosxdx??xcosx?sinx?C.
2. ?lnxdx;
解 ?lnxdx?xlnx??xdlnx?xlnx??dx?xlnx?x?C. 3. ?arcsinxdx;
解 ?arcsinxdx?xarcsinx??xdarcsinx ?xarcsinx??x1?x2dx
?xarcsinx?1?x2?C. 4. ?xe?xdx;
解 ?xe?xdx???xde?x??xe?x??e?xdx ??xe?x?e?x?C??e?x(x?1)?C. 5. ?x2lnxdx;
解 ?x2lnxdx??lnxdx3?x3lnx??x3dlnx
333 ?x3lnx??x2dx?x3lnx?x3?C.
3339 6. ?e?xcosxdx; 解 因为
?e?xcosxdx??e?xdsinx?e?xsinx??sinxde?x?e?xsinx??e?xsinxdx
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?e?xsinx??e?xdcosx?e?xsinx?e?xcosx??cosxde?x ?e?xsinx?e?xcosx??e?xcosxdx,
所以 ?e?xcosxdx?(e?xsinx?e?xcosx)?C?e?x(sinx?cosx)?C. 7. ?e?2xsindx; 解 因为
?e?2xsindx?2?e?2xdcos?2e?2xcos?2?cosde?2x
2222 ?2e?2xcos?4?e?2xcosdx?2e?2xcos?8?e?2xdsin 2222 ?2e?2xcos?8e?2xsin?8?sinde?2x 222 ?2e?2xcos?8e?2xsin?16?e?2xsindx, 222所以 ?e?2xsindx?? 8. ?xcosdx;
解 ?xcosdx?2?xdsin?2xsin?2?sindx?2xsin?4cos?C. 222222 9. ?x2arctanxdx;
解 ?x2arctanxdx??arctanxdx3?x3arctanx??x3?333111121212x2xxxxxxxxxxxxxxx22?2xxxe(cos?4sin)?C. 1722x2xxxxxx1?x131x2112132 ?xarctanx?? dx?xarctanx?(1?)dx?2361?x2361?x111 ?x3arctanx?x2?ln(1?x2)?C.
366dx
10. ?xtan2xdx
解 ?xtan2xdx??x(sec2x?1)dx??xsec2xdx??xdx??x2??xdtanx ??x2?xtanx??tanxdx??x2?xtanx?ln|cosx|?C.
仅供学习与交流,如有侵权请联系网站删除 谢谢5
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