What do I do when I have to integrate x multiplied by f(x)?

Question

Do I just find f(1)-f(0) and then multiply them by x? It doesn't seem right.

Edit: I have seemed to forget that integration by parts exists

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Answer 1 · 2017-05-04T05:05:52

The answer is B.

You cannot factor out an #x# out of the integral, you have to use integration by parts to simplify the integral.

Start out by solving the indefinite integral , then later use FTC to find the solution to the definite integral.
#int xf''(x) dx#

Use integration by parts: #color(blue)(int u dv = vu-int vdu)#

#color(green)(((u=x),(du=dx))((v=f'(x)),(dv=f''(x))))#

#int xf''(x) dx = xf'(x)-intf'(x)dx color(red)(+C)#

#=xf'(x)-f(x) color(red)(+C)#

Use the original bounds of definite integral to solve:
#int_0^1 x f''(x) dx = [xf'(x)-f(x)]_0^1#

#=[color(purple)(1)*f'(color(purple)(1))-f(color(purple)(1))]-[color(blue)(0)*f'(color(blue)(0))-f(color(blue)(0))]#

#=[1*6-2]-[0*(-2)-1]#

#=4+1#

#=5#

#"Answer choice "B#