设函数f(x)在[a,b]上有连续的导函数,且f(a)=f(b)=0,∫(b,a) [f(x)^2]dx=1,则∫(b,a) [xf(x)f'(x)]dx=?【注意∫()括号里面第一个数字是在上面,第二个数字在下面】(A)-1/2(B)1/2(C)0(D)1请问如何做
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/02 01:57:13
![设函数f(x)在[a,b]上有连续的导函数,且f(a)=f(b)=0,∫(b,a) [f(x)^2]dx=1,则∫(b,a) [xf(x)f'(x)]dx=?【注意∫()括号里面第一个数字是在上面,第二个数字在下面】(A)-1/2(B)1/2(C)0(D)1请问如何做](/uploads/image/z/14264453-29-3.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0f%28x%29%E5%9C%A8%5Ba%2Cb%5D%E4%B8%8A%E6%9C%89%E8%BF%9E%E7%BB%AD%E7%9A%84%E5%AF%BC%E5%87%BD%E6%95%B0%2C%E4%B8%94f%28a%29%3Df%28b%29%3D0%2C%E2%88%AB%28b%2Ca%29+%5Bf%28x%29%5E2%5Ddx%3D1%2C%E5%88%99%E2%88%AB%28b%2Ca%29+%5Bxf%28x%29f%27%28x%29%5Ddx%3D%3F%E3%80%90%E6%B3%A8%E6%84%8F%E2%88%AB%EF%BC%88%EF%BC%89%E6%8B%AC%E5%8F%B7%E9%87%8C%E9%9D%A2%E7%AC%AC%E4%B8%80%E4%B8%AA%E6%95%B0%E5%AD%97%E6%98%AF%E5%9C%A8%E4%B8%8A%E9%9D%A2%2C%E7%AC%AC%E4%BA%8C%E4%B8%AA%E6%95%B0%E5%AD%97%E5%9C%A8%E4%B8%8B%E9%9D%A2%E3%80%91%EF%BC%88A%EF%BC%89-1%2F2%EF%BC%88B%EF%BC%891%2F2%EF%BC%88C%EF%BC%890%EF%BC%88D%EF%BC%891%E8%AF%B7%E9%97%AE%E5%A6%82%E4%BD%95%E5%81%9A)
设函数f(x)在[a,b]上有连续的导函数,且f(a)=f(b)=0,∫(b,a) [f(x)^2]dx=1,则∫(b,a) [xf(x)f'(x)]dx=?【注意∫()括号里面第一个数字是在上面,第二个数字在下面】(A)-1/2(B)1/2(C)0(D)1请问如何做
设函数f(x)在[a,b]上有连续的导函数,且f(a)=f(b)=0,∫(b,a) [f(x)^2]dx=1,则∫(b,a) [xf(x)f'(x)]dx=?【注意∫()括号里面第一个数字是在上面,第二个数字在下面】
(A)-1/2
(B)1/2
(C)0
(D)1
请问如何做出这道题目,请尽可能通俗地,傻瓜式地讲解,要过程和你的最终选项,
设函数f(x)在[a,b]上有连续的导函数,且f(a)=f(b)=0,∫(b,a) [f(x)^2]dx=1,则∫(b,a) [xf(x)f'(x)]dx=?【注意∫()括号里面第一个数字是在上面,第二个数字在下面】(A)-1/2(B)1/2(C)0(D)1请问如何做
对“原式∫(b,a) [xf(x)f'(x)]dx”凑微分得
=∫(b,a) [xf(x)]df(x),用分部积分法得
=[xf(x)f(x)]代(b,a)-∫(b,a) [f(x)*(f(x)+xf'(x))]dx
=0-∫(b,a) f(x)f(x)dx-∫(b,a) [xf(x)f'(x)]dx= -1-∫(b,a) [xf(x)f'(x)]dx,最后一个积分是原式,得
原式=-1-原式,则原式=-1/2.
选A。考查分步积分。xf(x)f’(x)=1/2×x{f(x)^2}’,所以把f(x)f’(x)拿到d()里面,再打开后前面的含有f(a)和f(b),故为零,后面是题目已知的积分乘以1/2,由于前后两部分是差,故为-1/2,懂了吗,亲?
(B)