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: Differentiate the first option
Check by differentiating $\frac{1}{5}\log|x+2|-\frac{1}{10}\log(x^{2}+1)+\frac{2}{5}\tan^{-1}x$.

Step 2: Combine the derivative
$\frac{1}{5(x+2)}-\frac{x}{5(x^{2}+1)}+\frac{2}{5(x^{2}+1)}=\frac{1}{5(x+2)}+\frac{2-x}{5(x^{2}+1)}$.

Step 3: Simplify
Common denominator: $\frac{(x^{2}+1)+(2-x)(x+2)}{5(x+2)(x^{2}+1)}=\frac{x^{2}+1+4-x^{2}}{5(x+2)(x^{2}+1)}=\frac{1}{(x+2)(x^{2}+1)}$.
This matches the integrand.

Final Answer:
The first option differentiates back to the integrand, so it is correct. \[ \boxed{\text{(A) }\dfrac{1}{5}\log|x+2|-\dfrac{1}{10}\log(x^{2}+1)+\dfrac{2}{5}\tan^{-1}x+C} \]