理???0f(t)dt?F(0),
???0tf(t)dt??F'(0)在使用上面公式时应先考虑一下广义积分的存
在性.
例7. 计算积分?解:
??0??sinttdt.
???0sinttdt??0L[sint]ds????1s?120??0ds?arctanedt.
?ts??0??2
例8. 计算积分???0e?3tcos2tdt及??1?costt解: (1) L[cos2t]?ss?4??s2???0e?3tcos2tdt?L[cos2t]s?3?ss?412s22?s?3??313
(2) L[1?costt]???0?L[1?cost]ds?12???s(1s?ss?122)ds?lns?1?s12lns?1s22
令s?1, 得:
?1?costtedt??tln2.