evaluate the following integral\n int \frac{4 x^{2}+x+17}{(x+1)left(x^{2}+4\right)} d x \ncan partial…

evaluate the following integral\n int \frac{4 x^{2}+x+17}{(x+1)left(x^{2}+4\right)} d x \ncan partial fraction decomposition be used to evaluate the given integral? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. yes, partial fraction decomposition can be used. the given integral can be rewritten as ( int(quad) d x ), which is more readily evaluated.\nb. no, partial fraction decomposition cannot be used.

evaluate the following integral\n int \frac{4 x^{2}+x+17}{(x+1)left(x^{2}+4\right)} d x \ncan partial fraction decomposition be used to evaluate the given integral? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. yes, partial fraction decomposition can be used. the given integral can be rewritten as ( int(quad) d x ), which is more readily evaluated.\nb. no, partial fraction decomposition cannot be used.

Answer

Explanation:

Step1: Partial fraction decomposition condition

For a rational function (\frac{f(x)}{g(x)}) where (g(x)) can be factored into linear and irreducible quadratic factors, partial - fraction decomposition can be used. Here, the denominator is ((x + 1)(x^{2}+4)), which is factored into a linear factor ((x + 1)) and an irreducible quadratic factor ((x^{2}+4)).

Step2: Set up the partial - fraction decomposition

We assume (\frac{4x^{2}+x + 17}{(x + 1)(x^{2}+4)}=\frac{A}{x + 1}+\frac{Bx+C}{x^{2}+4}). Then (4x^{2}+x + 17=A(x^{2}+4)+(Bx + C)(x + 1)). Expanding the right - hand side: (4x^{2}+x + 17=(A + B)x^{2}+(B + C)x+(4A + C)). By equating the coefficients of the like - degree terms:

  • For (x^{2}): (A + B=4)
  • For (x): (B + C=1)
  • For the constant term: (4A + C=17) From (A + B=4), we have (B = 4 - A). Substitute (B = 4 - A) into (B + C=1), then (C=1-(4 - A)=A - 3). Substitute (C = A - 3) into (4A + C=17), we get (4A+(A - 3)=17), (5A=20), (A = 4). If (A = 4), then (B=0) and (C = 1). So (\frac{4x^{2}+x + 17}{(x + 1)(x^{2}+4)}=\frac{4}{x + 1}+\frac{1}{x^{2}+4}).

Answer:

A. Yes, partial fraction decomposition can be used. The given integral can be rewritten as (\int\left(\frac{4}{x + 1}+\frac{1}{x^{2}+4}\right)dx), which is more readily evaluated.