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

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

$\underset{2}{\overset{3}{\int }}udu=\frac{1}{2}{u}^2{|}_2^3=\frac{5}{2}$. This is our final answer.