I'm not certain on this but I did the following:

Dealing with each side of the equation individually:

[IMG]http://latex.codecogs.com/gif.latex?\displaystyle \begin{align*} \int f'(x)f''(x)dx \\ &= \int f''(x)dx \times f'(x)-\iint f''(x)dx \times Dx(f'(x)) \\ &= f'(x)^2 - \int f'(x)f''(x)dx \\ \\ \Rightarrow \int f'(x)f''(x)dx=\frac{1}{2}f'(x)^2 \end{align*}[/IMG]

And

[IMG]http://latex.codecogs.com/gif.latex?\begin{align*} a \int f'(x)f(x)^{0.4}dx \\ &=a\left ( \int f'(x)dx\times f(x)^{0.4}-\iint f'(x)dx \times Dx(f(x)^{0.4})dx \right ) \\ &=a\left ( f(x)^{1.4}-\int f(x) \times 0.4f'(x)f(x)^{-0.6}dx \right ) \\ &=a\left ( f(x)^{1.4}-0.4\int f'(x)f(x)^{0.4}dx \right ) \\ \Rightarrow a \int f'(x)f(x)^{0.4}dx = \frac{5a}{7}f(x)^{1.4} \end{align*}[/IMG]

Giving:

[IMG]http://latex.codecogs.com/gif.latex?\displaystyle \begin{align*} \frac{1}{2}f'(x)^2 = \frac{5a}{7}f(x)^{1.4} \end{align*}[/IMG]

Integrating one more time should be relatively straightforward now. But please don't work straight off this answer - Do everything above and make sure I haven't made a mistake - it's 2am right now. =D

EDIT: Sorry, I can't get the latex to display properly - just copy/paste everything between the image tags into your browser and you'll see the equations. If anyone can tell me why the equations aren't displaying I'd be grateful.