Question

If \(\int \frac{\left(x^4+1\right)}{x\left(x^2+1\right)^2} d x=A \log |x|+\frac{B}{1+x^2}+c\), then \(\mathrm{A}-\mathrm{B}\) is (where c is the constant of integration)

• A. 0
• B. 1
• C. 2
• D. -1

Hint:

Split the integrand into partial fractions, integrate, and compare with the given form to read A and B.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Write \(\dfrac{x^{4}+1}{x(x^{2}+1)^{2}}=\dfrac{a}{x}+\dfrac{bx+c}{x^{2}+1}+\dfrac{dx+e}{(x^{2}+1)^{2}}\).

Step 2: Find the constants
\(x^{4}+1=a(x^{2}+1)^{2}+(bx+c)x(x^{2}+1)+(dx+e)x\).
Comparing coefficients: \(a=1,\,b=0,\,c=0,\,d=-2,\,e=0\).

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

Step 4: Compare
So \(A=1\) and \(B=1\), giving \(A-B=0\).

Final Answer:
A - B equals 0. \[ \boxed{\text{(A) }0} \]