Question:

Step 1: Partial Fraction Decomposition The integrand is expressed in terms of partial fractions. $$ \frac{4x^3-7x^2+8x-4}{x^2(1+x^2)} = \frac{A}{x}+\frac{B}{x^2}+\frac{Cx+D}{1+x^2} $$ For the given integral, the partial fraction decomposition is: $$ \frac{4}{x}+\frac{7}{x^2}-\frac{4}{1+x^2} $$ Step 2: Integration Integrate the decomposed expression term by term. $$ \int\left(\frac{4}{x}+\frac{7}{x^2}-\frac{4}{1+x^2}\right)dx $$ $$ = 4\int\frac{1}{x}dx + 7\int x^{-2}dx - 4\int\frac{1}{1+x^2}dx $$ $$ = 4\ln|x| + 7\left(-\frac{1}{x}\right) - 4\arctan x + C $$ $$ = 4\ln|x| - \frac{7}{x} - 4\arctan x + C $$