作业帮指数函数、三角函数的乘积求积分
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/09 19:10:47
指数函数、三角函数的乘积求积分
两次分部积分,总是循环.请写出前几部来,提示思路.
两次分部积分,总是循环.请写出前几部来,提示思路.
原式=1/2m*1/4∫（0,π）sin3ade^2ma
=1/(8m)sin2a*e^(2ma)|(0,π)-1/(8m)∫（0,π）e^2madsin3a
=-3/(8m)∫（0,π）e^2ma*cos3ada
=-1/(2m)*3/(8m)∫（0,π）*cos3a*de^2ma
=-3/(16m²)*cos3a*e^2ma|（0,π）+3/(16m²)∫（0,π）e^2madcos3a
=3/(16m²)*e^2mπ+3/(16m²)-9/(16m²)∫（0,π）e^2ma*sin3ada
这时出现循环式了
设原式=x
x=3/(16m²)*e^2mπ+3/(16m²)-9/(16m²)x
(1+9/(16m²))x=3/(16m²)*e^2mπ+3/(16m²)
两边同乘以16m²,得
(16m²+9)x=3e^2mπ+3
x=(3e^2mπ+3)/(16m²+9)
=1/(8m)sin2a*e^(2ma)|(0,π)-1/(8m)∫（0,π）e^2madsin3a
=-3/(8m)∫（0,π）e^2ma*cos3ada
=-1/(2m)*3/(8m)∫（0,π）*cos3a*de^2ma
=-3/(16m²)*cos3a*e^2ma|（0,π）+3/(16m²)∫（0,π）e^2madcos3a
=3/(16m²)*e^2mπ+3/(16m²)-9/(16m²)∫（0,π）e^2ma*sin3ada
这时出现循环式了
设原式=x
x=3/(16m²)*e^2mπ+3/(16m²)-9/(16m²)x
(1+9/(16m²))x=3/(16m²)*e^2mπ+3/(16m²)
两边同乘以16m²,得
(16m²+9)x=3e^2mπ+3
x=(3e^2mπ+3)/(16m²+9)