DOCUMENTO

1. Demetrio Ccesa Rayme
Integración por Fracciones Parciales

2. Ejemplo 1
න
2𝑥2
+ 3
𝑥2 + 1 2
𝑑𝑥

3. 2𝑥2+3
𝑥2+1 2 =
𝐴𝑥+𝐵
𝑥2+1
+
𝐶𝑥+𝐷
𝑥2+1 2
2𝑥2+3
𝑥2+1 2 =
𝐴𝑥3+𝐴𝑥+𝐵𝑥2+𝐵+𝐶𝑥+𝐷
𝑥2+1 2
න
2𝑥2 + 3
𝑥2 + 1 2 𝑑𝑥
2𝑥2
+ 3 = 𝐴𝑥3
+ 𝐴𝑥 + 𝐵𝑥2
+ 𝐵 + 𝐶𝑥 + 𝐷

4. 2𝑥2
+ 3 = 𝐴𝑥3
+ 𝐵𝑥2
+ 𝐴 + 𝐶 𝑥 + 𝐵 + 𝐷
𝐴 = 0 𝐵 = 2
𝐴 + 𝐶 = 0
𝐵 + 𝐷 = 3
→ 0 + 𝐶 = 0 → 𝐶 = 0
→ 2 + 𝐷 = 3 → 𝐷 = 1

5. 2𝑥2+3
𝑥2+1 2 =
𝐴𝑥+𝐵
𝑥2+1
+
𝐶𝑥+𝐷
𝑥2+1 2
2𝑥2+3
𝑥2+1 2 =
2
𝑥2+1
+
1
𝑥2+1 2
න
2𝑥2
+ 3
𝑥2 + 1 2 𝑑𝑥 = න
2𝑑𝑥
𝑥2 + 1
+ න
𝑑𝑥
𝑥2 + 1 2
𝐴 = 0
𝐵 = 2
𝐶 = 0
𝐷 = 1

6. න
2𝑥2
+ 3
𝑥2 + 1 2
𝑑𝑥 = න
2𝑑𝑥
𝑥2 + 1
+ න
𝑑𝑥
𝑥2 + 1 2
Integral Original:
න
2𝑑𝑥
𝑥2 + 1
= 2 න
𝑑𝑥
𝑥2 + 1
tan 𝑧 =
𝑥
1
𝑑𝑥 = sec2
𝑧 𝑑𝑧
tan 𝑧 = 𝑥
sec 𝑧 =
𝑥2 + 1
1
sec 𝑧 = 𝑥2 + 1
sec2
𝑧 = 𝑥2
+ 1
z
1
x

7. 𝑑𝑥 = sec2 𝑧 𝑑𝑧
sec2 𝑧 = 𝑥2 + 1
tan 𝑧 = 𝑥
න
2𝑑𝑥
𝑥2 + 1
= 2 න
𝑑𝑥
𝑥2 + 1
න
2𝑑𝑥
𝑥2 + 1
= 2 tan−1 𝑥 + 𝐶
= 2 න
sec2
𝑧 𝑑𝑧
sec2 𝑧
= 2 න 𝑑𝑧 = 2𝑧 + C

8. 𝑑𝑥 = sec2 𝑧 𝑑𝑧
sec2 𝑧 = 𝑥2 + 1
න
𝑑𝑥
𝑥2 + 1 2
= න
sec2
𝑧 𝑑𝑧
sec2 𝑧 2
= න
1
sec2 𝑧
𝑑𝑧
න
2𝑥2
+ 3
𝑥2 + 1 2
𝑑𝑥 = න
2𝑑𝑥
𝑥2 + 1
+ න
𝑑𝑥
𝑥2 + 1 2
Integral Original:

9. sec 𝑥 =
1
cos 𝑥
Identidad Trigonométrica:
→ cos 𝑥 =
1
sec 𝑥
න
1
sec2 𝑧
𝑑𝑧 = න cos2
𝑧 𝑑𝑧
= න cos 𝑧 ∙ cos 𝑧 𝑑𝑧
𝑢 = cos 𝑧
𝑑𝑢 = − sin 𝑧 𝑑𝑧
𝑑𝑣 = cos 𝑧 𝑑𝑧
න 𝑑𝑣 = න cos 𝑧 𝑑𝑧
𝑣 = sin 𝑧
= cos 𝑧 ∙ sin 𝑧 − න sin 𝑧 − sin 𝑧 𝑑𝑧
= cos 𝑧 ∙ sin 𝑧 + න sin2 𝑧 𝑑𝑧

10. Identidad Trigonométrica:
sin2 𝑥 = 1 − cos2 𝑥න cos2
𝑧 𝑑𝑧 = cos 𝑧 ∙ sin 𝑧 + න(1 − cos2
z) 𝑑𝑧
න cos2
𝑧 𝑑𝑧 = cos 𝑧 ∙ sin 𝑧 + න 𝑑𝑧 − න cos2
z 𝑑𝑧
න cos2
𝑧 𝑑𝑧 = cos 𝑧 ∙ sin 𝑧 + 𝑧 − න cos2
z 𝑑𝑧
2 න cos2
𝑧 𝑑𝑧 = cos 𝑧 ∙ sin 𝑧 + 𝑧
න cos2
𝑧 𝑑𝑧 =
1
2
(cos 𝑧 ∙ sin 𝑧) +
1
2
𝑧 + 𝐶

11. න cos2
𝑧 𝑑𝑧 =
1
2
(cos 𝑧 ∙ sin 𝑧) +
1
2
𝑧 + 𝐶
=
1
2
𝑥
𝑥2 + 1
+
1
2
tan−1
𝑥 + 𝐶
z
1
x
tan 𝑧 = 𝑥
=
1
2
1
𝑥2 + 1
∙
𝑥
𝑥2 + 1
+
1
2
tan−1
𝑥 + 𝐶
න
𝑑𝑥
𝑥2 + 1 2
=
1
2
𝑥
𝑥2 + 1
+
1
2
tan−1
𝑥 + 𝐶

12. න
2𝑥2
+ 3
𝑥2 + 1 2
𝑑𝑥 = න
2𝑑𝑥
𝑥2 + 1
+ න
𝑑𝑥
𝑥2 + 1 2
න
𝑑𝑥
𝑥2 + 1 2 =
1
2
𝑥
𝑥2 + 1
+
1
2
tan−1 𝑥 + 𝐶 න
2𝑑𝑥
𝑥2 + 1
= 2 tan−1 𝑥 + 𝐶
න
2𝑥2
+ 3
𝑥2 + 1 2
𝑑𝑥 = 2 tan−1
𝑥 +
1
2
𝑥
𝑥2 + 1
+
1
2
tan−1
𝑥 + 𝐶
න
2𝑥2
+ 3
𝑥2 + 1 2
𝑑𝑥 =
5
2
tan−1
𝑥 +
1
2
𝑥
𝑥2 + 1
+ 𝐶

13. Ejemplo 2
• Resolver ‫׬‬
sin 𝑥 𝑑𝑥
cos 𝑥 1+cos2 𝑥
, sea cos 𝑥 = 𝑢

14. 𝑢 = cos 𝑥
𝑥 = cos−1
𝑢
𝑑𝑥 = −
1
1 − 𝑢2
𝑑𝑢
Identidad Trigonométrica:
sin2 𝑥 + cos2 𝑥 = 1
sin2
𝑥 + 𝑢2
= 1
sin 𝑥 = 1 − 𝑢2
න
sin 𝑥 𝑑𝑥
cos 𝑥 1 + cos2 𝑥
= න
1 − 𝑢2
𝑢 1 + 𝑢2
∙ −
1
1 − 𝑢2
𝑑𝑢
Fórmula:
𝑑
𝑑𝑥
tan−1 𝑣 = −
𝑑𝑣
𝑑𝑥
1 − 𝑣2

15. න
1 − 𝑢2
𝑢 1 + 𝑢2
∙ −
1
1 − 𝑢2
𝑑𝑢 = − න
1
𝑢 1 + 𝑢2
1
𝑢 1 + 𝑢2
=
𝐴
𝑢
+
𝐵𝑢 + 𝐶
1 + 𝑢2
1
𝑢 1 + 𝑢2
=
𝐴 + 𝐴𝑢2
+ 𝐵𝑢2
+ 𝐶𝑢
𝑢 1 + 𝑢2
1 = 𝐴 + 𝐵 𝑢2
+ 𝐶𝑢 + 𝐴
𝐶 = 0
𝐴 = 1
𝐴 + 𝐵 = 0 → 𝐵 = −1
− න
1
𝑢 1 + 𝑢2
= − න
1
𝑢
𝑑𝑢 + න
−1𝑢
1 + 𝑢2
𝑑𝑢

16. − න
𝑑𝑢
𝑢
− න
𝑢 𝑑𝑢
1 + 𝑢2
= න
𝑢 𝑑𝑢
1 + 𝑢2
− න
𝑑𝑢
𝑢
=
1
2
ln 1 + 𝑢2
− ln 𝑢 + 𝐶
= ln 1 + 𝑢2
1
2 − ln 𝑢 + 𝐶
=
1
2
න
2𝑢 𝑑𝑢
1 + 𝑢2
− න
𝑑𝑢
𝑢

17. න
sin 𝑥 𝑑𝑥
cos 𝑥 1 + cos2 𝑥
= ln
1 + cos2 𝑥
cos 𝑥
+ 𝐶
Integral Original:
න
sin 𝑥 𝑑𝑥
cos 𝑥 1 + cos2 𝑥
cos 𝑥 = 𝑢
− න
1
𝑢 1 + 𝑢2
= ln
1 + 𝑢2
ln 𝑢
+ 𝐶

18. Ejemplo 3
න
2 + tan2
𝜃 sec2
𝜃 𝑑𝜃
1 + tan3 𝜃

19. න
2 + tan2
𝜃 sec2
𝜃 𝑑𝜃
1 + tan3 𝜃
𝑢 = tan 𝜃
Identidad Trigonométrica:
1 + tan2 𝜃 = sec2 𝜃
1 + u2
= sec2
𝜃
𝜃 = tan−1
𝑢
𝑑𝜃 =
1
1 + u2
𝑑𝑢
= න
2 + u2
(1 + u2
)
1 + u3
∙
1
1 + u2
𝑑𝑢
= න
2 + u2
𝑑𝑢
1 + u3
= න
2 + u2
𝑑𝑢
(1 + 𝑢) 1 − 𝑢 + 𝑢2

20. 2 + u2
(1 + 𝑢) 1 − 𝑢 + 𝑢2
=
𝐴
1 + 𝑢
+
𝐵𝑢 + 𝐶
1 − 𝑢 + 𝑢2
=
𝐴 + 𝐵 𝑢2
+ −𝐴 + 𝐵 + 𝐶 𝑢 + 𝐴 + 𝐶
(1 + 𝑢)(1 − 𝑢 + 𝑢2)
2 + u2
= 𝐴 + 𝐵 𝑢2
+ −𝐴 + 𝐵 + 𝐶 𝑢 + 𝐴 + 𝐶
𝐴 + 𝐵 = 1
−𝐴 + 𝐵 + 𝐶 = 0
𝐴 + 𝐶 = 2
𝐴 = 1
𝐵 = 0
𝐶 = 1

21. 𝐴 = 1
𝐵 = 0
𝐶 = 1
න
2 + u2
𝑑𝑢
(1 + 𝑢) 1 − 𝑢 + 𝑢2
= න
𝐴
1 + 𝑢
𝑑𝑢 + න
𝐵𝑢 + 𝐶
1 − 𝑢 + 𝑢2
𝑑𝑢
= න
1
1 + 𝑢
𝑑𝑢 + න
𝑑𝑢
1 − 𝑢 + 𝑢2
= ln(1 + 𝑢) + න
𝑑𝑢
𝑢2 − 𝑢 + 1
4
+
3
4
= ln(1 + 𝑢) + න
𝑑𝑢
𝑢 − 1
2
2
+ 3
2
2

22. z
3
2
𝑢 − 1
2
tan 𝑧 =
2𝑢 − 1
3
𝑢 =
3 tan 𝑧 + 1
2
𝑑𝑢 =
3
2
sec2
𝑑𝑧
sec 𝑧 =
2 ∙ 𝑢 − 1
2
2
+ 3
2
2
3
3
4
𝑠𝑒𝑐2
𝑧 = 𝑢 − 1
2
2
+ 3
2
2

23. 𝑑𝑢 =
3
2
sec2
𝑑𝑧
3
4
sec2
𝑧 = 𝑢 − 1
2
2
+ 3
2
2
𝑧 = tan−1
2𝑢 − 1
3
න
𝑑𝑢
𝑢 − 1
2
2
+ 3
2
2 = න
3
2
sec2
𝑑𝑧
3
4
sec2 𝑧
=
2 3
3
න 𝑑𝑧 =
2 3
3
𝑧 + 𝐶
න
2 + u2
𝑑𝑢
(1 + 𝑢) 1 − 𝑢 + 𝑢2
= ln(1 + 𝑢) + න
𝑑𝑢
𝑢 − 1
2
2
+ 3
2
2
න
2 + u2
𝑑𝑢
(1 + 𝑢) 1 − 𝑢 + 𝑢2
= ln(1 + 𝑢) +
2 3
3
tan−1
2𝑢 − 1
3
+ 𝐶

24. Integral Original:
න
2 + tan2
𝜃 sec2
𝜃 𝑑𝜃
1 + tan3 𝜃
𝑢 = tan 𝜃
න
2 + u2
𝑑𝑢
(1 + 𝑢) 1 − 𝑢 + 𝑢2
= ln(1 + 𝑢) +
2 3
3
tan−1
2𝑢 − 1
3
+ 𝐶
න
2 + tan2
𝜃 sec2
𝜃 𝑑𝜃
1 + tan3 𝜃
= ln 1 + tan 𝜃 +
2 3
3
tan−1
2 tan 𝜃 − 1
3

25. Ejemplo 4
න
𝑡 + 3
𝑡2 + 4𝑡 + 5
2
𝑑𝑡

26. න
𝑡 + 3
𝑡2 + 4𝑡 + 5
2
𝑑𝑡 = න
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
𝑑𝑡
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
=
𝐴𝑡 + 𝐵
𝑡2 + 4𝑡 + 5
+
𝐶𝑡 + 𝐷
𝑡2 + 4𝑡 + 5 2
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
=
(𝐴𝑡 + 𝐵) 𝑡2
+ 4𝑡 + 5 + 𝐶𝑡 + 𝐷
𝑡2 + 4𝑡 + 5 2
𝑡2
+ 6𝑡 + 9 = 𝐴𝑡3
+ 4𝐴𝑡2
+ 5𝐴𝑡 + 𝐵𝑡2
+ 4𝐵𝑡 + 5𝐵 + 𝐶𝑡 + 𝐷

27. 𝑡2
+ 6𝑡 + 9 = 𝐴𝑡3
+ 4𝐴 + 𝐵 𝑡2
+ 5𝐴 + 4𝐵 + 𝐶 𝑡 + (5𝐵 + 𝐷)
𝐴 = 0
4𝐴 + 𝐵 = 1 → 𝐵 = 1
5𝐴 + 4𝐵 + 𝐶 = 6→
𝐷 = 45𝐵 + 𝐷 = 9 →
𝐶 = 2
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
=
𝐴𝑡 + 𝐵
𝑡2 + 4𝑡 + 5
+
𝐶𝑡 + 𝐷
𝑡2 + 4𝑡 + 5 2
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
=
1
𝑡2 + 4𝑡 + 5
+
2𝑡 + 4
𝑡2 + 4𝑡 + 5 2

28. න
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
𝑑𝑡 = න
𝑑𝑡
𝑡2 + 4𝑡 + 5
+ න
(2𝑡 + 4)
𝑡2 + 4𝑡 + 5 2
𝑑𝑡
= න
𝑑𝑡
𝑡 + 2 2 + 1
+ න(2𝑡 + 4) 𝑡2
+ 4𝑡 + 5 −2
𝑑𝑡
= න
𝑑𝑡
𝑡 + 2 2 + 1
+
𝑡2
+ 4𝑡 + 5 −1
−1
= න
𝑑𝑡
𝑡 + 2 2 + 1
−
1
𝑡2 + 4𝑡 + 5

29. න
𝑑𝑡
𝑡 + 2 2 + 1
z
1
𝑡 + 2
tan 𝑧 =
𝑡 + 2
1
𝑑𝑡 = sec2
𝑧 𝑑𝑧
tan 𝑧 = 𝑡 + 2
sec 𝑧 =
𝑡 + 2 2 + 1
1
sec 𝑧 = (𝑡 + 2)2+1
sec2
𝑧 = (𝑡 + 2)2
+1
𝑡 = tan 𝑧 − 2

30. 𝑑𝑡 = sec2 𝑧 𝑑𝑧
sec2 𝑧 = (𝑡 + 2)2+1
tan 𝑧 = 𝑡 + 2
න
𝑑𝑡
𝑡 + 2 2 + 1
= න
sec2
𝑧 𝑑𝑧
sec2 𝑧
= න 𝑑𝑧 = 𝑧 + 𝐶
න
𝑑𝑡
𝑡 + 2 2 + 1
= tan−1
𝑡 + 2 + 𝐶

31. න
𝑑𝑡
𝑡 + 2 2 + 1
= tan−1 𝑡 + 2 + 𝐶
න
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
𝑑𝑡 = න
𝑑𝑡
𝑡 + 2 2 + 1
−
1
𝑡2 + 4𝑡 + 5
න
𝑡2
+ 6𝑡 + 9
𝑡2 + 4𝑡 + 5 2
𝑑𝑡 = tan−1
𝑡 + 2 −
1
𝑡2 + 4𝑡 + 5
+ 𝐶