Science:Math Exam Resources/Courses/MATH105/April 2014/Question 01 (f)/Solution 3

We use integration by substitution. If $u=f'(x)$ then $du=f''(x)dx$. Thus, $f'(x)f''(x)dx=udu$. We also can transform the bounds so that $1\mapsto f'(1)=2$ and $2\mapsto f'(2)=3$. In terms of u, we can integrate:

$\int _{2}^{3}udu={\frac {1}{2}}u^{2}|_{2}^{3}={\frac {5}{2}}$. This is our final answer.