Here is what I've done so far...

so

=

=

Now we have to find what the coefficients are equal to, so

A+C=1

-4A+B=-4

B=0

A=1

B=0

C=0

So now I should be able to take the integral of

This equals on the interval [7,8], so my answer should be .

This is apparently the wrong answer, so I must have made a mistake somewhere, but I'm not seeing it.