Question

x f(x) f'(x)
0 2 5
4 -3 11
Consider the twice-differentiable function \(f\) that has values of \(f\) and its derivative for x=0 and x=4 listed in the table above.
\[If~~\int\limits_{0}^{4} f(x)~dx = 8\]what is the value of \[\int\limits_{0}^{4} x~f'(x)~dx~~~?\]

Answer 1

a) -20
b) -13
c) -12
d) -7
e) 36

Answer 2

do integration bye parts :-
\(\int\limits_{0}^{4} x~f'(x)~dx = xf(x)|_0^4 - \int_0^4 f(x) dx\)

Answer 3

This is what I've tried so far...
integration by parts
∫xf'(x) dx
u = x v = f(x)
du = dx dv = f'(x) dx
uv - ∫v du
(x)(f(x)) - ∫f'(x) dx

4(-3) - 8
-12 - 8
-20 ?

Answer 4

ur correct :)

Answer 5

Thank you! :D

Answer 6

but be carfull in the integral its f(x) not f'(x )
(x)(f(x)) - ∫f(x) dx

Answer 7

np :)

Answer 8

ah right typo :P

Answer 9

it should be like this :- ( for net time )
u = x dv = f(x) dx
du = dx v = f(x)

Answer 10

next*

Answer 11

alright, I'll keep that in mind ^_^

Answer 12

^_^