evaluate the following integral\n\\( \\int \\frac { 7 ( x ^ { 2 } + 4 ) } { x ( x ^ { 2 } + 7 ) } d x…

evaluate the following integral\n\\( \\int \\frac { 7 ( x ^ { 2 } + 4 ) } { x ( x ^ { 2 } + 7 ) } 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 ( \\square ) d x \\), which is more readily evaluated.\nb. no, partial fraction decomposition cannot be used.

evaluate the following integral\n\\( \\int \\frac { 7 ( x ^ { 2 } + 4 ) } { x ( x ^ { 2 } + 7 ) } 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 ( \\square ) d x \\), which is more readily evaluated.\nb. no, partial fraction decomposition cannot be used.

Answer

Explanation:

Step1: Set up partial fraction decomposition

Let (\frac{7(x^{2}+4)}{x(x^{2}+7)}=\frac{A}{x}+\frac{Bx + C}{x^{2}+7}). Then (7(x^{2}+4)=A(x^{2}+7)+(Bx + C)x). Expanding gives (7x^{2}+28=(A + B)x^{2}+Cx+7A).

Step2: Solve for (A), (B), and (C)

Equating coefficients: For (x^{2}): (7 = A + B). For (x): (0 = C). For the constant term: (28 = 7A), so (A = 4). Substituting (A = 4) into (7 = A + B) gives (B = 3). So (\frac{7(x^{2}+4)}{x(x^{2}+7)}=\frac{4}{x}+\frac{3x}{x^{2}+7}).

Step3: Integrate

(\int\frac{7(x^{2}+4)}{x(x^{2}+7)}dx=\int(\frac{4}{x}+\frac{3x}{x^{2}+7})dx). (\int\frac{4}{x}dx+\int\frac{3x}{x^{2}+7}dx). For (\int\frac{4}{x}dx = 4\ln|x|). For (\int\frac{3x}{x^{2}+7}dx), let (u = x^{2}+7), (du = 2xdx), so (\int\frac{3x}{x^{2}+7}dx=\frac{3}{2}\ln|x^{2}+7|).

Answer:

A. Yes, partial fraction decomposition can be used. The given integral can be rewritten as (\int(\frac{4}{x}+\frac{3x}{x^{2}+7})dx), which is more readily evaluated.