大师作文网求下面几道题的积分,大家帮下忙,求详解,
来源：学生作业帮 编辑：大师作文网作业帮 分类：数学作业 时间：2024/11/18 02:03:24
求下面几道题的积分,大家帮下忙,求详解,
1.原式=∫cos^2(x)sin^2(x)d(sinx)=∫(1-sin^2(x))sin^2(x)d(sinx)=∫sin^2(x)d(sinx)-∫sin^4(x)d(sinx)=1/3sin^3(x)-1/5sin^5(x)+C
2.原式=2∫d(√x)/(1+(√x)^2)=2arctan√x+C
3.令√x=t,则原式=2∫tsintdt=-2∫td(cost)=-2tcost+2∫costdt=-2tcost+2sint+C=-2√xcos√x+2sin√x+C
4.原式=∫(x-1-2)/((x-1)^2+1)dx=∫(x-1)d(x-1)/((x-1)^2+1)-2∫d(x-1)/((x-1)^2+1)=1/2∫d((x-1)^2+1)/((x-1)^2+1)-2∫d(x-1)/((x-1)^2+1)=1/2ln((x-1)^2+1)-2arctan(x-1)+C
2.原式=2∫d(√x)/(1+(√x)^2)=2arctan√x+C
3.令√x=t,则原式=2∫tsintdt=-2∫td(cost)=-2tcost+2∫costdt=-2tcost+2sint+C=-2√xcos√x+2sin√x+C
4.原式=∫(x-1-2)/((x-1)^2+1)dx=∫(x-1)d(x-1)/((x-1)^2+1)-2∫d(x-1)/((x-1)^2+1)=1/2∫d((x-1)^2+1)/((x-1)^2+1)-2∫d(x-1)/((x-1)^2+1)=1/2ln((x-1)^2+1)-2arctan(x-1)+C