Question

\(\int \frac{dx}{(x+2)(x^{2}+1)}\) is equal to

• A. \(\frac{1}{5}\log |x+2|-\frac{1}{10}\log (x^{2}+1)+\frac{2}{5}\tan ^{-1}x+C\)
• B. \(\log |x+2|+\frac{1}{10}\log (x^{2}+1)+\frac{1}{5}\tan ^{-1}x+C\)
• C. \(5\,\log |x+2|+10\,\log (x^{2}+1)+\tan ^{-1}x+C\)
• D. \(\log |x+2|+\frac{1}{10}\log (x^{2}+1)-\frac{1}{5}\tan ^{-1}x+C\)

Hint:

Split the integrand into partial fractions with a linear and a quadratic factor, then integrate each term.

Answer 1

The Correct Option is A

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

Step 2: Find the constants
\(1=A(x^{2}+1)+(Bx+C)(x+2)\). Putting \(x=-2\): \(1=5A\), so \(A=\frac{1}{5}\).
Comparing \(x^{2}\): \(A+B=0\Rightarrow B=-\frac{1}{5}\). Constant term: \(A+2C=1\Rightarrow C=\frac{2}{5}\).

Step 3: Integrate term by term
\(\int\frac{dx}{(x+2)(x^{2}+1)}=\frac{1}{5}\log|x+2|-\frac{1}{10}\int\frac{2x}{x^{2}+1}dx+\frac{2}{5}\int\frac{dx}{x^{2}+1}\).

Step 4: Result
\(=\frac{1}{5}\log|x+2|-\frac{1}{10}\log(x^{2}+1)+\frac{2}{5}\tan^{-1}x+C\).

Final Answer:
The integral is (1/5)log|x+2| − (1/10)log(x²+1) + (2/5)tan⁻¹x + C. \[ \boxed{\text{(A) }\dfrac{1}{5}\log|x+2|-\dfrac{1}{10}\log(x^{2}+1)+\dfrac{2}{5}\tan^{-1}x+C} \]