Question

If \( \displaystyle \int \frac{dx}{(x+2)(x^2+1)} = p\log|x+2| + q\log|x^2+1| + r\tan^{-1}x + c \), then \( p+q+r = \)

• A. \( \frac{2}{5} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{7}{10} \)
• D. 16

Hint:

For rational functions with linear and quadratic factors, use partial fractions before integration.

Answer 1

The Correct Option is B

Step 1: Use partial fractions.
\[ \frac{1}{(x+2)(x^2+1)} = \frac{A}{x+2}+\frac{Bx+C}{x^2+1}. \]

Step 2: Multiply by denominator.
\[ 1=A(x^2+1)+(Bx+C)(x+2). \]

Step 3: Expand.
\[ 1=Ax^2+A+Bx^2+2Bx+Cx+2C. \]
\[ 1=(A+B)x^2+(2B+C)x+(A+2C). \]

Step 4: Compare coefficients.
\[ A+B=0,\quad 2B+C=0,\quad A+2C=1. \]

Step 5: Solve for constants.
From \(A+B=0\),
\[ B=-A. \]
From \(2B+C=0\),
\[ C=-2B=2A. \]
Now,
\[ A+2C=1. \]
\[ A+4A=1. \]
\[ 5A=1 \Rightarrow A=\frac{1}{5}. \]
So,
\[ B=-\frac{1}{5},\quad C=\frac{2}{5}. \]

Step 6: Integrate.
\[ \int \frac{dx}{(x+2)(x^2+1)} = \int \left[\frac{1}{5(x+2)}+\frac{-\frac{x}{5}+\frac{2}{5}}{x^2+1}\right]dx. \]
\[ = \frac{1}{5}\log|x+2|-\frac{1}{10}\log|x^2+1|+\frac{2}{5}\tan^{-1}x+c. \]

Step 7: Identify \(p,q,r\).
\[ p=\frac{1}{5},\quad q=-\frac{1}{10},\quad r=\frac{2}{5}. \]
Therefore,
\[ p+q+r=\frac{1}{5}-\frac{1}{10}+\frac{2}{5} = \frac{2-1+4}{10} = \frac{5}{10} = \frac{1}{2}. \]
Final Answer:
\[ \boxed{\frac{1}{2}} \]