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Consider the following integral \(\int \frac{5x^3 + 2x + 1}{x^4 + x^2} dx\).

1. Which of the following is the correct partial sums decomposition?

a.  \(\frac{A}{x} + \frac{Bx}{x^2} + \frac{Cx + D}{x^2 + 1}\)

b.  \(\frac{Ax + B}{x^2} + \frac{Cx + D}{x^2 + 1}\)

c.  \(\frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 1}\)

d.  \(\frac{A}{x} + \frac{Bx + C}{x^2} + \frac{Cx}{x^2 + 1}\)

e.  \(\frac{A}{x} + \frac{Bx + C}{x^2} + \frac{C}{x^2 + 1}\)

2. Solve for \(A\) and \(B\).

a.  \(A = 1, B = 2\)

b.  \(A = 2, B = 1\)

c.  \(A = 1, B = 0\)

d.  \(A = 0, B = 1\)

e.  \(A = 0, B = 2\)

3. Solve for \(C\) and \(D\).

a.  \(C = 3, D = -1\)

b.  \(C = -2, D = 1\)

c.  \(C = -2, D = 0\)

d.  \(C = 2, D = 0\)

e.  \(C = 0, D = 0\)

4. Find the value of the original integral.

a.  \(\frac{\frac{5}{4}x^4+ x^2 + x}{\frac{1}{5}x^5 + \frac {1}{3}x^3} + c\)

b.  \(\frac{1}{x}-\ln{x^2 + 1} + c\)

c.  \(\ln\mid x\mid - 2\ln(x^2 + 1) + c\)

d.  \(\ln\mid x\mid + 3\ln(x^2 + 1) + c\)

e.  \(2 \ln \mid x \mid - \frac{1}{x} + \frac{3}{2}\ln(x^2+1) - \arctan x + c\)