Question

$\int \frac{dx}{x(x^3 + 1)} = \dots$

• A. $\log \left(\frac{x^3}{x^3 + 1}\right) + c$
• B. $\frac{1}{3} \log \sqrt{\frac{x^3 + 1}{x^3}} + c$
• C. $\log \sqrt{\frac{x^3 + 1}{x^3}} + c$
• D. $\frac{1}{3} \log \left(\frac{x^3}{x^3 + 1}\right) + c$

Hint:

Memorize this highly recurring integral shortcut: $\int \frac{dx}{x(x^n + 1)} = \frac{1}{n} \log \left| \frac{x^n}{x^n + 1} \right| + c$. You can apply it instantly to save time.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To integrate fractions with $x(x^n+1)$, multiply the numerator and denominator by $x^{n-1}$ to facilitate substitution.

Step 2: Formula Application:
Multiply by $x^2$: $I = \int \frac{x^2 \, dx}{x^3(x^3+1)}$. Let $t = x^3 \implies dt = 3x^2 \, dx \implies \frac{1}{3} dt = x^2 \, dx$.

Step 3: Explanation:
$I = \frac{1}{3} \int \frac{dt}{t(t+1)} = \frac{1}{3} \int \left(\frac{1}{t} - \frac{1}{t+1}\right) dt$. $I = \frac{1}{3} [\log |t| - \log |t+1|] + c = \frac{1}{3} \log \left|\frac{t}{t+1}\right| + c$. $I = \frac{1}{3} \log \left(\frac{x^3}{x^3 + 1}\right) + c$.

Step 4: Final Answer:
The integral is $\frac{1}{3} \log \left(\frac{x^3}{x^3 + 1}\right) + c$.