I have been trying to solve an integral but am having no luck. I would appreciate help. I have tried substitutions, partial fractions. I just dont see how to do it.
integral (u4+1)du/(u5+3u).
I would appreciate any help.
To add to what Subhotosh Khan said, when you realise that \(\displaystyle \displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}u} \,\left( u^5 + 3\,u \right) = 5\,u^4 + 3 \end{align*}\), notice that you ALMOST have that in your numerator, but you need to do some manipulation...
\(\displaystyle \displaystyle \begin{align*} \int{ \frac{u^4 + 1}{u^5 + 3\,u} \,\mathrm{d}u } &= \frac{1}{5} \int{ \frac{5\,u^4 + 5}{u^5 + 3\,u} \,\mathrm{d}u } \\ &= \frac{1}{5} \int{ \left( \frac{5\,u^4 + 3}{u^5 + 3\,u} + \frac{2}{u^5 + 3\,u} \right) \,\mathrm{d}u } \\ &= \frac{1}{5} \int{ \frac{5\,u^4 + 3}{u^5 + 3\,u} \,\mathrm{d}u } + \frac{2}{5} \int{ \frac{1}{u^5 + 3\,u} \,\mathrm{d}u } \end{align*}\)
The first of those integrals is solved with a substitution, and the second needs partial fractions.
