Question

\(\int \frac{\mathrm{d} x}{x\left(x^3+1\right)}=\)

• A. \(\mathrm{log}(\frac{x^{3}}{x^{3}+1})+\mathrm{c}\), where c is the constant of integration
• B. \(\frac{1}{3}\mathrm{log}(\sqrt[\frac{x^{3}}{x^{3}+1}]{3})+\mathrm{c}\), where c is the constant of integration
• C. \(\mathrm{log}(\sqrt[\frac{x^{3}}{x^{3}+1}]{3})+\mathrm{c}\), where c is the constant of integration
• D. \(\frac{1}{3}\mathrm{log}(\frac{x^{3}+1}{x^{3}})+\mathrm{c}\), where c is the constant of integration

Hint:

Multiply the numerator and denominator by x squared and substitute t = x cubed.

Answer 1

The Correct Option is C

Step 1: Formula-first
$\int\dfrac{dt}{t(t+1)}=\log\left|\dfrac{t}{t+1}\right|$.

Step 2: Substitution
With $t=x^3$ the factor $\dfrac13$ appears from $x^2dx=\dfrac{dt}{3}$.

Step 3: Elimination
Option (A) lacks the $\dfrac13$, (D) has the fraction inverted, and (B) carries an extra $\dfrac13$. Option (C) has exactly one factor $\dfrac13$, as a cube root.

Final Answer:
The answer is one-third of the log of the ratio x cubed over (x cubed + 1). \[ \boxed{\text{(C) }\log\left(\dfrac{x^{3}}{x^{3}+1}\right)^{\frac13}+c} \]