Assignment 14  :  Hints and Solutionsp  .576

2. \(TraditionalForm
      \`\(5  x - 7\)\/\(x\^2 - 3  x\  + \ 2\)\  = 
        \(\(5  x - 7\)\/\(\((x - 1)\) \((x - 2)\)\) = \ 
          A\/\(x - 1\) + \ B\/\(x - 2\)\)\)

Apart[((5  x - 7)/((x^2) - 3  x  +  2))]

8. \(TraditionalForm
      \`\(t\^4 + 9\)\/\(t\^4 + 9  t\^2\) = 
        \(1 - \(9  t\^2 - \ 9\)\/\(\(t\^2\)(t\^2 + 9)\) = \ 
          1 + \ A\/t + \ B\/t\^2 + \ \(C\ t\  + \ D\)\/\(t\^2 + 9\)\)\)

Apart[(((t^4) + 9)/((t^4) + 9  (t^2)))]

10. (1/((x^2) + x))  =  
        (1/(x (x + 1))) =  (A/x) +  (B/(x + 1))

Apart[(1/((x^2) + x))]

16. \(TraditionalForm
      \`\(x + 3\)\/\(2  x\^3 - \ 8  x\) = 
        \(\(x + 3\)\/\(2 \( x(x - 2)\) \((x + 2)\)\) = \ 
          A\/x + B\/\(x - 2\) + \ C\/\(x + 2\)\)\)

Apart[((x + 3)/(2  (x^3) -  8  x))]

18. ((x^3)/((x^2) - 2  x + 1)) = 
        x + 2  +  ((3  x - 2)/(((x - 1))^2)) = 
          x + 2 + (A/(x - 1)) +  (B/(((x - 1))^2))

Apart[((x^3)/((x^2) - 2  x + 1))]

22. ((3  (t^2) +  t  +  4)/((t^3) + t)) = 
        ((3  (t^2) +  t  +  4)/(t ((t^2) + 1))) =  
          (A/t) + ((B t  +  C)/((t^2) + 1))

Apart[((3  (t^2) +  t  +  4)/((t^3) + t))]

26. (((s^4) +  81)/(s ((((s^2) +  9))^2)))  = 
        (A/s) + ((B s  +  C)/((s^2) +  9)) +  
          ((D s  +  E)/((((s^2) +  9))^2))

Apart[(((s^4) +  81)/(s ((((s^2) +  9))^2)))]

30. ((x^4)/(((x^2) - 1)))  =  
        (x^2) + 1 + (1/((x - 1) (x + 1))) =  
          (x^2) + 1 + (A/(x - 1)) + (B/(x + 1))

Apart[((x^4)/(((x^2) - 1)))]

34. \(TraditionalForm
      \`\(\(2  y\^4\)\/\(y\^3 - \ y\^2 + \ y - 1\) = 
        \(2  y + 2 + 2\/\(\(y\^2\)(y - 1) + \ y - 1\) = 
          \(2  y + 2 + 2\/\(\((y - 1)\) \((y\^2 + 1)\)\) = 
            2  y + 2\  + \ A\/\(y - 1\) + \ \(B\ y + C\)\/\(y\^2 + 1\)\)\)\ 
      \)\)

Apart[((2  (y^4))/((y^3) -  (y^2) +  y - 1))]

38. Substitute u  =  cos(θ), \(TraditionalForm
      \`du\  = \ 
        \(-\(sin(θ)\)\) dθ\): \(TraditionalForm
      \`\(\(sin(θ)\) 
            dθ\)\/\(\(cos\^2\)(θ)\  + \ 
            cos(θ)\  - \ 2\) = 
        \(-\ du\)\/\(u\^2 + \ u - \ 2\)\);
 \(TraditionalForm
      \`\(\ 1\)\/\(u\^2 + \ u - \ 2\) = \ 
        \(1\/\(\((u - 1)\) \((u + 2)\)\) = \ 
          A\/\(u - 1\) + B\/\(u + 2\)\)\)

Apart[((- 1)/((u^2) +  u -  2))]

51. You can use Mathematica to integrate this one.

(∫_0^1)(( (x^4) (((x - 1))^4))/((x^2) + 1)) 
        \[DifferentialD]x

Using partial fractions: (((x^4) (((x - 1))^4))/((x^2) + 1)) = 
        (((x^8) - 4  (x^7) + 6  (x^6) - 4  (x^5) +  (x^4))/((x^2) + 1)) = 
          (x^6) - 4  (x^5) + 5  (x^4) - 4  (x^2) + 4  - (4/((x^2) + 1))

Apart[(((x^4) (((x - 1))^4))/((x^2) + 1))]

∫(4 - 4 (x^2) + 5 (x^4) - 4 (x^5) + (x^6) - (4/(1 + (x^2))))
           \[DifferentialD]x

I'll leave plotting the function (((x^4) (((x - 1))^4))/((x^2) + 1)) for you to experiment with.