What is the integrer of f(x)

[MethMath11]

Diketahui means : the datas are (sorry for my bad translation)

 

[mmm4444bot]

For tutors (from translate.google.com):

"Jika diketahui" means "If known"

"dan" means "and"

"maka" means "then"

?

 

[MarkFL]

I didn't see where IBP helps, so I constructed the following function:

[MATH]f(x)=-9x^2+10x+1[/MATH]
We see that it meets all requirements, yet:

[MATH]\int_0^1 xf(x)\,dx=\frac{19}{12}[/MATH]
Interestingly, I get the same result if I use:

[MATH]f(x)=\begin{cases}7x+1, & 0\le x<\dfrac{1}{2} \\[3pt] -5x+7, & \dfrac{1}{2}\le x\le1 \\ \end{cases}[/MATH]

 

[MarkFL]

I've revisited this problem several times today, and I'm wondering if there is a typo in the problem, and we are actually supposed to find:

[MATH]\int_0^1 xf'(x)\,dx[/MATH]
IBP then tells us, where:

[MATH]u=x\implies du=dx[/MATH]
[MATH]dv=f'(x)\,dx\implies v=f(x)[/MATH]
[MATH]\int_0^1 xf'(x)\,dx=\left[xf(x)\right]_0^1-\int_0^1 f(x)\,dx=2-3=-1[/MATH]
This is actually one of the given choices.