Fracciones parciales

74. Fracciones parciales

Ejercicio 75.1

Calcule las siguientes integrales indefinidas

1: $\int \frac{1}{x^{4} - x^{2}} 𝑑𝑥$
$\frac{1}{x} + \frac{1}{2} \ln | \frac{x - 1}{x + 1} | + C .$
2: $\int \frac{3 x^{2} - 4 x + 5}{(x - 1) (x^{2} + 1)} 𝑑𝑥$
$2 \ln | x - 1 | + \frac{1}{2} \ln (x^{2} + 1) - 3 \arctan x + C .$
3: $\int \frac{p^{3} + 14 p^{2} + 2}{4 p^{4} + 4 p^{3} - 7 p^{2} + 2 p} 𝑑𝑝$
$- \frac{9}{4 (2 p - 1)} + \frac{1}{4} \ln | 2 p - 1 | + \ln | p | - \ln | p + 2 | + C .$
4: $\int \frac{5 z^{3} - z^{2} + 4 z + 4}{(z^{2} + z) (z^{3} - z)} 𝑑𝑧$
$\frac{4}{z} - \frac{3}{z + 1} - 3 \ln | z + 1 | + 3 \ln | z - 1 | + C .$
5: $\int \frac{26 u - 34 + 2 u^{2}}{{(u - 2)}^{3} {(2 u + 1)}^{2}} 𝑑𝑢$
$- \frac{318}{625} \ln | u - 2 | - \frac{66}{125 (u - 2)} - \frac{13}{25 {(u - 2)}^{2}} + \frac{318}{625} \ln | 2 u + 1 | - \frac{186}{125 (2 u + 1)} + C .$
6: $\int \frac{u - 8}{u^{2} - 2 u - 8} 𝑑𝑢$
$\frac{5}{3} \ln | u + 2 | - \frac{2}{3} \ln | u - 4 | + C .$
7: $\int \frac{8 - 10 w^{2} - w^{3}}{8 w + 2 w^{2} - w^{3}} 𝑑𝑤$
$w + \ln | w | + 2 \ln | w + 2 | + 9 \ln | w - 4 | + C .$
8: $\int \frac{- x^{2} + x - 1}{(4 - x^{2}) {(x - 5)}^{2}} 𝑑𝑥$
$\frac{\ln | x - 2 |}{12} - \frac{\ln | x + 2 |}{28} - \frac{\ln | x - 5 |}{21} - \frac{1}{x - 5} + C .$
9: $\int \frac{9^{x} + 3^{x} + 6}{9^{x} - 3^{x} - 2} 𝑑𝑥$
$2 \log_{3} | 3^{x} - 2 | + 2 \log_{3} | 3^{x} + 1 | - 3 x + C .$
10: $\int \frac{x^{2} + 1}{x^{2} - x} 𝑑𝑥$
$x - \ln | x | + 2 \ln | x - 1 | + C .$
11: $\int \frac{1}{{(x + 5)}^{2} (x - 1)} 𝑑𝑥$
$- \frac{1}{36} \ln | x + 5 | + \frac{1}{6 (x + 5)} + \frac{1}{36} \ln | x - 1 | + C .$
12: $\int \frac{1}{x^{3} - 1} 𝑑𝑥$
$\frac{\ln | x - 1 |}{3} - \frac{\ln | x^{2} + x + 1 |}{6} - \frac{\arctan (\frac{2 x + 1}{\sqrt{3}})}{\sqrt{3}} + C .$
13: $\int \frac{5 x^{2} + 3 x + 12}{x^{3} + 4 x} 𝑑𝑥$
$3 \ln | x | + \frac{3}{2} \arctan (\frac{x}{2}) + \ln | x^{2} + 4 | + C .$
14: $\int \frac{5 x^{2} + 12 x + 1}{x^{3} + 3 x^{2} - 4} 𝑑𝑥$
$2 \ln | x - 1 | + 3 \ln | x + 2 | - \frac{1}{x + 2} + C .$
15: $\int \frac{x + 3}{2 x + 1} 𝑑𝑥$
$\frac{5}{4} \ln | 2 x + 1 | + \frac{x}{2} + C .$
16: $\int \frac{2 x^{2} + 4 x + 5}{x - 1} 𝑑𝑥$
$11 \ln | x - 1 | + x^{2} + 6 x + C .$

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