Question

\[ \int \frac{\mathrm{d} x}{x\left(x^2+1\right)}= \]

• A. \(\mathrm{log}(x)-\frac{1}{2}\mathrm{log}(x^{2}+1)+\mathrm{c}\), where c is the constant of integration.
• B. \(\frac{1}{2}\mathrm{log}(x)-\mathrm{log}(x^{2}+1)+\mathrm{c}\), where c is the constant of integration.
• C. \(\mathrm{log}(x)+\frac{1}{2}\mathrm{log}(x^{2}+1)+\mathrm{c}\), where c is the constant of integration.
• D. \(-\mathrm{log}(x)-\frac{1}{2}\mathrm{log}(x^{2}+1)+\mathrm{c}\), where c is the constant of integration.

Hint:

Split the fraction into partial fractions and integrate each part.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Write \(\dfrac{1}{x(x^{2}+1)}=\dfrac{A}{x}+\dfrac{Bx+C}{x^{2}+1}\).

Step 2: Find the Constants
\(1=A(x^{2}+1)+(Bx+C)x\). Put \(x=0\): \(A=1\).
Comparing \(x^{2}\): \(A+B=0\), so \(B=-1\); comparing \(x\): \(C=0\).

Step 3: Integrate
\(\int\left(\dfrac{1}{x}-\dfrac{x}{x^{2}+1}\right)dx=\log x-\dfrac{1}{2}\log(x^{2}+1)+c\).

Step 4: Match
This is option (A).

Final Answer:
The integral is \(\log x-\dfrac{1}{2}\log(x^{2}+1)+c\). \[ \boxed{\text{(A) }\log x-\dfrac{1}{2}\log\left(x^{2}+1\right)+c} \]