C1 2017 1 Edexcel

1

It helps to write:

\displaystyle 2x^5 - \frac 1 {4x^3} - 5 = 2x^5 - \frac 1 4 x^{-3} - 5

Then:

\begin{align*}\int \left(2x^5 - \frac 1 {4x^3}\right) \mathrm dx & = \frac {2x^{5 + 1}} {5 + 1} - \frac 1 4 \times \frac {x^{-3 + 1}} {-3 + 1} - 5x + C \\ & = \frac 1 3 x^6 + \frac 1 8 x^{-2} - 5x + C\end{align*}