What is $\int \frac{x}{x^5-7} dx$?

Question

I have tried out many trigonometric substitution like $x=\sin^{\frac{2}{5}}z$. But it did not work.

Answer 1

Substitute $x=\sqrt[5]{7}u$, and you have $$ \begin{align} \int\frac{\sqrt[5]{7}u}{7u^5-7}\,\sqrt[5]{7}du &=\frac{\sqrt[5]{49}}{7}\int\frac{u}{u^5-1}\,du\\ &=\frac{\sqrt[5]{49}}{7}\int\left(\frac{A}{u-1}+\frac{Bu+C}{u^2-2hu+1}+\frac{Du+E}{u^2-2ku+1}\right)\,du\\ \end{align} $$

where $h=\cos(2\pi/5)=\frac{\sqrt{5}-1}{4}$ and $k=\cos(4\pi/5)=\frac{-\sqrt{5}-1}{4}$. The values of $A,B,C,D,E$ can be found using linear algebra, after recombining the three terms and comparing the coefficients of powers of $u$ in the numerator to that of $\frac{u}{u^5-1}$.

Continuing: $$ \begin{align} &=\frac{\sqrt[5]{49}}{7}\int\left(\frac{A}{u-1}+\frac{Bu+C}{(u-h)^2+1-h^2}+\frac{Du+E}{(u-k)^2+1-k^2}\right)\,du\\ &=\frac{\sqrt[5]{49}}{7}\int\left(\frac{A}{u-1}+\frac{B(u-h)+Bh+C}{(u-h)^2+1-h^2}+\frac{D(u-k)+Dk+E}{(u-k)^2+1-k^2}\right)\,du\\ &=\frac{\sqrt[5]{49}}{7}\left(A\ln\lvert u-1\rvert+\frac{B}{2}\ln\left((u-h)^2+1-h^2\right)+\frac{Bh+C}{\sqrt{1-h^2}}\arctan\left(\frac{u-h}{\sqrt{1-h^2}}\right)\right.\\ &\phantom{{}={}}\left.+\frac{D}{2}\ln\left((u-k)^2+1-k^2\right)+\frac{Dh+E}{\sqrt{1-k^2}}\arctan\left(\frac{u-k}{\sqrt{1-k^2}}\right)\right)+\text{constant}\\ \end{align} $$ Note $1-h^2=\sin^2(2\pi/5)$ and $1-k^2=\sin^2(4\pi/5)$, which may help some of you needed to carry this through to an explicit end.

The "constant" may take two different values on either side of $u=1$ where the discontinuity is.

The last step would be to back-substitute from $u$ to $x$. Then the "constant" might change values at $x=\sqrt[5]{7}$.

Answer 2

To make the problem more general, consider $$I_n=\int\frac x {x^n-A}\,dx=A^{\frac 2n-1}\int\frac t {t^n-1}\,dt=A^{\frac 2n-1}\int\frac t {\prod_{i=1}^n(t-r_i)}\,dt$$ where $r_i$ are the roots of unity. Using partial fraction decomposition, you will end with $$I_n=A^{\frac 2n-1}\sum _{i=1}^n\int \frac {\alpha_i}{t-r_i}\,dt=A^{\frac 2n-1}\sum _{i=1}^n{\alpha_i}\log({t-r_i})$$ For sure, since the ${\alpha_i},r_i$ terms will be complex numbers, you will need to recombine them by pairs if you want to get rid of all complex terms (coefficients and logarithms).

There is also a shorter notation $$J_n=\int\frac t {t^n-1}\,dt=-\frac{1}{2} t^2 \, _2F_1\left(1,\frac{2}{n};1+\frac{2}{n};t^n\right)$$ where appears the Gaussian or ordinary hypergeometric function.