Integrating with Multiple Variables: How to Solve the Crazy Integration Problem?

[Spartan029]

Homework Statement

Let $$f$$ be twice differentiable with $$f(0)=6, f(1)=5,$$ $$f'(1)=2$$
Evaluate the integral $$\int_0^{1}x f''(x) dx$$

Homework Equations

$$\int uv' dx = uv = \int u'v dx$$

The Attempt at a Solution

u = x and v' = f''(x)
so
u' = 1dx and v = f'(x)
so

$$xf'(x) - \int_0^{1}f'(x) dx$$

from here I am not sure what to do... maybe parts again...
u = 1dx and v' = f'(x)
so
u' = 0? and v = f(x) //derivative of 1dx is 0 right?
so

$$xf'(x) - f(x) - \int_0^{1} 0 dx$$ ?

i really don't know. any help would be awesome!

 

[gabbagabbahey]

What does the fundamental theorem of calculus say about:

$$\int_0^1 f'(x)dx$$

?

 

[Spartan029]

$$\int_0^1 f'(x)dx = f(1) - f(0)$$ ??

 

[gabbagabbahey]

Yep.

 

[Spartan029]

wow that's awesome. now i feel retarded haha.
but seriosuly... thanks gabba!