- \(\displaystyle \int \frac {x^3 + 4x^2}{x^2 + 4x + 3} ~dx = \color{blue} {\frac {1}{2} x^2 - \frac {9}{2} \ln (x + 3) + \frac {3}{2} \ln (x+1)}\)
- Solution: \(\frac {x^3 + 4x^2}{x^2 + 4x + 3} = x - \frac {9}{2} \cdot \frac {1}{x + 3} + \frac {3}{2} \cdot \frac {1}{x + 1}\)
- \(\displaystyle \int \frac {2x^3 + 2x^2 - 9x - 1}{x^2 + x - 6} ~dx = \color{blue} {x^2 +2 \ln (x + 3) + \ln (x - 2)}\)
Solution: \(\frac {2x^3 + 2x^2 - 9x - 1}{x^2 + x - 6} = 2x + 2 \cdot \frac {1}{x + 3} + \frac {1}{x - 2}\)
Used 2019, Unsolved
- \(\displaystyle \int \frac {x^3 - x^2 -3x +1}{x^2 - x - 6} ~dx = \color{blue} {\frac {1}{2} x^2 + 2 \ln (x - 3) + \ln (x + 2)}\)
- Solution: \(\frac {x^3 - x^2 -3x +1}{x^2 - x - 6} = x + 2 \cdot \frac {1}{x - 3} + \frac {1}{x + 2}\)
- \(\displaystyle \int \frac {3x^3 - 12x^2 + 15x - 5}{x^2 - 4x + 4} ~dx = \color{blue} {\frac {3}{2} x^2 + 3 \ln (x - 2) - \frac {1}{x - 2}}\)
- Solution: \(\frac {3x^3 - 12x^2 + 15x - 5}{x^2 - 4x + 4} = 3x + 3 \cdot \frac {1}{x - 2} + \frac {1}{(x - 2)^2}\)
- \(\displaystyle \int \frac {x(x^2 - 3x + 5)}{x^2 - 2x + 1} ~dx = \color{blue} {\frac {1}{2} x^2 - x + \ln (x - 1) - \frac {3}{x - 1}}\)
- Solution: \(\frac {x(x^2 - 3x + 5)}{x^2 - 2x + 1} = x - 1 + 2 \cdot \frac {1}{x - 1} + 3 \cdot \frac {1}{(x - 1)^2}\)
- \(\displaystyle \int \frac {x^3 -4x^2 + 2x - 3}{x+2} ~dx = \color{blue} {\frac 1 3 x^3 -3x^2 + 14x - 31\ln(x+2)}\)
Solution: \(\frac {x^3 - 4x^2 + 2x - 3}{x+2} = x^2 - 6x + 14 - 31 \cdot \frac {1}{x + 2}\)
Used 2018, Unsolved
- \(\displaystyle \int \frac {x^4 - 2x^3 - 8x^2 + 2x + 10 }{x^2 - 2x - 8} ~dx = \color{blue} {\frac {1}{3} x^3 + 3 \ln (x - 4) - \ln (x + 2)}\)
- Solution: \(\frac {x^4 - 2x^3 - 8x^2 + 2x + 10 }{x^2 - 2x - 8} = x^2 + 3 \cdot \frac {1}{x - 4} - \frac {1}{x + 2}\)
Solution
1. \(\int \frac{x^3 + 4x^2}{x^2 + 4x + 3} \, dx = \frac{1}{2}x^2 + \frac{3}{2}\ln|x+1| - \frac{9}{2}\ln|x+3| + C\).
1.1. Long division: \(x^3 + 4x^2 = x(x^2 + 4x + 3) - 3x\), so \(\frac{x^3 + 4x^2}{x^2 + 4x + 3} = x - \frac{3x}{(x+1)(x+3)}\).
1.2. Partial fractions: \(-\frac{3x}{(x+1)(x+3)} = \frac{A}{x+1} + \frac{B}{x+3} \implies -3x = A(x+3) + B(x+1)\).Setting \(x = -1 \implies 3 = 2A \implies A = 3/2\).Setting \(x = -3 \implies 9 = -2B \implies B = -9/2\).
1.3. Integrating: \(\int \left(x + \frac{3/2}{x+1} - \frac{9/2}{x+3}\right) dx = \frac{1}{2}x^2 + \frac{3}{2}\ln|x+1| - \frac{9}{2}\ln|x+3| + C\).
2. \(\int \frac{2x^3 + 2x^2 - 9x - 1}{x^2 + x - 6} \, dx = x^2 + 2\ln|x+3| + \ln|x-2| + C\).
2.1. Long division: \(2x^3 + 2x^2 - 9x - 1 = 2x(x^2 + x - 6) + (3x - 1)\).
2.2. Partial fractions: \(\frac{3x-1}{(x+3)(x-2)} = \frac{A}{x+3} + \frac{B}{x-2} \implies 3x - 1 = A(x-2) + B(x+3)\).Setting \(x = 2 \implies 5 = 5B \implies B = 1\).Setting \(x = -3 \implies -10 = -5A \implies A = 2\).
2.3. Integrating: \(\int \left(2x + \frac{2}{x+3} + \frac{1}{x-2}\right) dx = x^2 + 2\ln|x+3| + \ln|x-2| + C\).
3. \(\int \frac{3x^3 - 12x^2 + 15x - 5}{(x-2)^2} \, dx = \frac{3}{2}x^2 + 3\ln|x-2| - \frac{1}{x-2} + C\).
3.1. Substitute \(u = x - 2 \implies x = u + 2, dx = du\).
3.2. \(3(u+2)^3 - 12(u+2)^2 + 15(u+2) - 5 = 3(u^3 + 6u^2 + 12u + 8) - 12(u^2 + 4u + 4) + 15(u+2) - 5 = 3u^3 + 6u^2 + 3u + 1\).
3.3. Dividing by \(u^2\): \(3u + 6 + \frac{3}{u} + \frac{1}{u^2}\).
3.4. Integrating: \(\frac{3}{2}u^2 + 6u + 3\ln|u| - \frac{1}{u} + C = \frac{3}{2}(x-2)^2 + 6(x-2) + 3\ln|x-2| - \frac{1}{x-2} + C = \frac{3}{2}x^2 + 3\ln|x-2| - \frac{1}{x-2} + C'\).