Evaluate the integral: I = integral (x^8-1)/(x (x^8+1)) dx

Evaluate the integral: $$ I = \int \frac{x^8-1}{x\left(x^8+1\right)}\,dx. $$

Step 1 · Rewrite and split

$$ \frac{x^8-1}{x(x^8+1)} = \frac1x - \frac{2}{x(x^8+1)} = -\frac1x + \frac{2x^7}{x^8+1}, $$

using $\dfrac{1}{x(x^8+1)}=\dfrac1x-\dfrac{x^7}{x^8+1}$.

Step 2 · Integrate each piece

$$ I = -\int \frac{dx}{x} + 2\int \frac{x^7}{x^8+1}\,dx = -\ln|x| + \frac14\ln\left(x^8+1\right) + C. $$

Summary Table

Additional Notes

• Correction: a source draft of this problem wrote the numerator as "$x^s-1$" (with the denominator still fixed at $x^8+1$) and attempted to generalize the integral to arbitrary $s$ using a series expansion, arriving at an unwieldy hypergeometric-function expression. Since the accompanying denominator is fixed at the 8th power throughout, and this problem is otherwise identical to Problem 264, the "$s$" is almost certainly a typo for "$8$" — in which case the integral is entirely elementary (no special functions needed), as shown above. If a genuinely different exponent $s\neq8$ was intended, that would need to be specified as a separate, distinct problem.
• The domain excludes $x=0$, where the denominator vanishes.
• The result can be checked by differentiating $-\ln|x|+\tfrac14\ln(x^8+1)$: this gives $-\tfrac1x+\tfrac{2x^7}{x^8+1}$, and combining over $x(x^8+1)$ gives numerator $-(x^8+1)+2x^8=x^8-1$, matching the integrand exactly.