很简单的数学题 integration by parts, 不知道上一步到下一步怎么做的

谢谢啦！！

I guess you missed some information in part a). Also, the first expression is an equation, while the second one is not. Hence, there is some typo here as well.
The first missing piece of information should be "B is the upper limit of the support of the distribution function, i.e., F(B)=1", which is the reason that you see (1-\beta)*w_2\bar in the second expression.
The second missing piece of information should be the condition of "(1-\beta )c=beta(B-w_2\bar)".

是不是还有（1-beta）＊C阿？

F（B）^2=1吗？分部积分的结果，还差一些东西，楼主是不是漏了什么，后面只是一个表达式而不是方程。