f. \( \int \frac{x^{3}+6 x-7}{\left(x^{2}-2 x-3\right)^{2}} \)

To solve the integral \( \int \frac{x^{3}+6 x-7}{\left(x^{2}-2 x-3\right)^{2}} \, dx \), first, factor the denominator. The expression \( x^{2}-2 x-3 \) can be factored as \( (x-3)(x+1) \). This means the integral can be expressed as: \[ \int \frac{x^{3}+6 x-7}{\left((x-3)(x+1)\right)^{2}} \, dx \] Next, to perform partial fraction decomposition, express \( \frac{x^{3}+6 x-7}{(x-3)^{2}(x+1)^{2}} \) as: \[ \frac{A}{x-3} + \frac{B}{(x-3)^{2}} + \frac{C}{x+1} + \frac{D}{(x+1)^{2}} \] Multiply through by the common denominator \( (x-3)^{2}(x+1)^{2} \) and equate coefficients to solve for \( A, B, C, \) and \( D \). After finding the constants, you can integrate each term separately. Be prepared for logarithmic forms and arctangent functions depending on the final form after decomposing. Finally, you should arrive at a solution that includes integration of the logarithm and potential polynomial terms. Don’t forget the arbitrary constant of integration at the end! Happy integrating!