Science:Math Exam Resources/Courses/MATH105/April 2014/Question 01 (f)/Solution 2

We use integration by parts and set $u=f^{\prime }(x)$ and $dv=f^{″}(x)dx$. Then

$\underset{1}{\overset{2}{\int }}{f}^{\prime }(x)f^{″}(x)dx={[f^{\prime }(x{)}^2]}_1^2-\underset{1}{\overset{2}{\int }}{f}^{″}(x)f^{\prime }(x)dx$

Note that the integral on the left and right are the same and thus we can rewrite this as

$2\underset{1}{\overset{2}{\int }}{f}^{\prime }(x)f^{″}(x)dx={[f^{\prime }(x{)}^2]}_1^2=3^2-2^2=9-4=5$

Dividing by 2 leads to $\underset{1}{\overset{2}{\int }}{f}^{\prime }(x)f^{″}(x)dx=\frac{5}{2}$