Ejercicios de antiderivadas

Históricamente la idea de integral se halla unida al cálculo de áreas a través del teorema fundamental del cálculo. Ampliamente puede decirse que la integral contiene información de tipo general mientras que la derivada la contiene de tipo local.
El concepto operativo de integral se basa en una operación contraria a la derivada a tal razón se debe su nombre de: antiderivada.
Las reglas de la derivación son la base que de cada operación de integral indefinida o antiderivada.

1. Alumno. Jesús Alan López Montañez
Carrera. Procesos industriales
Sección. 4 “A”
Prof. Gerardo Edgar Mata Ortiz

2. Introducción
Históricamente la idea de integral se halla unida al cálculo de áreas a
través del teorema fundamental del cálculo. Ampliamente puede decirse
que la integral contiene información de tipo general mientras que la
derivada la contiene de tipo local.
El concepto operativo de integral se basa en una operación contraria a
la derivada a tal razón se debe su nombre de: antiderivada.
Las reglas de la derivación son la base que de cada operación de
integral indefinida o antiderivada.

3. Formula 1 ∫�??????=??????+�
1-∫3??????
4
�??????=3∫??????
4
dx
=3
??????
4
5
+�
=3??????
5
+�

2-∫5??????
2
�??????=5∫??????
2
�??????
=5
??????
3
3
+�
=3??????
3
+�

3-∫
7
??????
3
�??????=7∫??????
2
�??????
=7∫
??????
−2
−2
+�
=
−1
??????
2⁄+�

4-∫
10
??????
3
�??????=10∫??????
−4
�??????
=10∫
??????
−3
−3
+c
=
−2
??????
3⁄+�

5-∫30??????
5
�??????=30∫??????
5
�??????
=30
??????
6
6
+�
=6??????
6
+�

4. Formula 2 ∫??????
??????
�??????=
??????
??????+1
??????+1
+�
1-∫10??????
4
−2sec??????)�??????= 10∫??????
4
�??????−2∫sec??????�??????
= 10
??????
5
5
−2tan??????+�
= 2??????
5
−2tan??????+�
2-∫
1
??????
4
�??????= ∫dx4dx
= �∫
1
??????
4
�??????
= �∫??????−4�??????
= � (−13??????−3+�)
=
−1
3??????
3
�+�
3-∫(13−??????)�??????= ∫13�??????+∫−??????�??????
= 13??????+� ∫−??????�??????
= 13??????+�−(12??????2+�)
= 13??????−12??????2+�

4-∫??????
2
�??????= ∫??????
2+1
2+1
�??????
=
??????
3
3
+�

5-∫(20−??????)�??????= ∫20�??????+∫−??????�??????
= 20??????+� ∫−??????�??????
= 20??????+�−(19??????2+�)
= 20??????−19??????2+�

5. Formula 3 ∫ (du + dv – dw) = ∫du + ∫dv - ∫dw

1-∫ (2+3??????
2
−8??????
3
)�??????=∫2�??????+ ∫3??????
2
�??????+ ∫−8??????
3

= 2??????+�+ ∫3??????
2
�?????? ∫−8??????
3
�??????
= 2??????+�+3 ∫??????
2
�??????+∫−8??????
3
�??????
= 2??????+�+3 (
1
3
??????
3
+�)+∫−8??????
3
�??????
= 2??????+�+3 (
??????
3
3
+�)+∫−8??????
3
�??????
= 2??????+�+3(
??????
3
3
+�) ∫−8∫??????
3
�??????
= 2??????+�+3 (
??????
3
3
+�)−8 (
1
4
??????
4
+�)
= 2??????+�+3 (
??????
3
3
+�)−8(
??????
4
4
+�)
= 2??????+??????
3
−8
??????
4
4
+�
= 2??????+??????
3
+
−8
1

??????
4
4
+�
= 2??????+??????
3
+
−8??????
4
4
+�
= 2??????+??????
3
−2??????
4
+�

6. 2-∫ (4??????
3
-3??????
2
+6??????−1)dx = ∫ 4??????
3
�??????+∫−3??????
2
�??????+∫−1�??????
= 4∫4??????
3
�??????+∫−3??????
2
�??????+ ∫6??????�??????+ ∫−1�??????
= 4 (
1
4
??????4
1
+�)+∫−3??????
2
�??????+∫6??????�??????+ ∫−1�??????
= 4(
??????
4
4
+�)+ ∫−3??????
2
�??????+ ∫6??????�??????+ ∫−1�??????
= 4(
??????
4
4
+�)+ −3∫??????
2
�??????+ ∫6??????�??????+ ∫−1�??????
= 4(
??????
4
4
+�)+ −3∫(
1
3
??????
3
+�) ∫6??????�??????+ ∫−1�??????
= 4(
??????
4
4
+�)+ −3∫(
??????
3
3
+ �)∫6??????�??????+ ∫−1�??????
= 4(
??????
4
4
+�)+ −3∫(
??????
3
3
+ �)+6∫??????�??????+ ∫−1�??????
= 4(
??????
4
4
+�)+ −3∫(
??????
3
3
+ �)+6∫(
1
2
??????
2
+ �) ∫−1�??????
= 4(
??????
4
4
+�)−3(
??????
3
3
+�)+6(
??????
2
2
+�)∫−1�??????
= 4(
??????
4
4
+�)−3(
??????
3
3
+�)+6(
??????
2
2
+�)+(−??????+2)
=??????
4
−??????
3
+3??????
2
−??????+�

7. 3-∫(2??????
2
−4??????+4)= ∫2??????
2
�??????+∫−4??????�??????+∫4�??????
= 2∫??????
2
�??????+∫−4??????�??????+∫4�??????
= 2∫
1
3
??????
3
+�∫−4??????�??????+∫4�??????
= 2 (
??????
3
3
+�)+∫−4�??????+∫4�??????
= 2 (
??????
3
3
+�)−4(
1
2
??????
2
+�)+∫4�??????
= 2 (
??????
3
3
+�)−4(
??????
2
2
+�)+∫4�??????
= 2 (
??????
3
3
+�)−4(
??????
2
2
+�)+(4??????+�)
=
2??????
3
3
−2??????
2
+4??????+�

8. 4-∫ (3??????
5
−2??????
3
) dx = ∫3??????
5
�??????+∫−2??????
3
�??????
= 3∫??????
5
�??????+∫−2??????
3
�??????
= 3 (
1
6
??????
6
+�)+∫−2??????
3
�??????
= 3 (
1
6
??????
6
+�)−2∫??????
3
�??????
= 3 (
1
6
??????
6
+�)−2 (
1
4
??????
4
+�)
=
??????
6
2
+2
??????
4
4
+�
=
??????
6
2
+
2(−??????
4
)
2(2)
+�
=
??????
6
2
+
−??????
4
)
2
+�
=
??????
6
2
−
−??????
4
)
2
+�
=
1
2
??????
6
+(
1
2
??????
4
+�)
=
1
2
??????
6
−
1
2
??????
4
+�

9. 5-∫ (3−2??????+??????
2
) dx = ∫3�??????+∫−2??????�??????+ ∫??????
2
�??????
= 3??????+�∫−2??????�??????+ ∫??????
2
�??????
= 3??????+�−2∫??????�??????+ ∫??????
2
�??????
= 3??????+�−2 (
1
2
??????
2
+�)+ ∫??????
2
�??????
= 3??????+�−2 (
??????
2
2
+�)+ ∫??????
2
�??????
= 3??????+�−2 (
??????
2
2
+�)+(
1
3
??????
3
+�)
= 3??????+�−2 (
??????
2
2
+�)+
??????
3
3
+�
= 3??????−??????
2
+
??????
3
3
+�
= 3??????−??????
2
+
1
3
??????
3
+�

10. Formula 4 ∫??????�??????=?????? ∫�??????
1-∫18??????
4
�??????= 18 ∫??????
4
�??????
= 18(
1
5
??????
5
+�)
= 18(
??????
5
5
+�)
=
18
1
??????
5
5
+�
=
18
1
??????
5
+�

2-∫20??????
2
�??????= 20∫??????
2
�??????
= 20(
1
3
??????
3
+�)
= 20(
??????
3
3
+�)
=
20
1
??????
3
3
+�
=
20
3
??????
3
+�

3-∫15??????
3
�??????=15∫??????
3
�??????
= 15(
1
3
??????
4
+�)
= 15(
??????
4
4
+�)
=
15
1
??????
4
4
+�
=
15
4
??????
4
+�

11. 4-∫50??????
5
�??????= 50∫??????
5
�??????
= 50(
1
6
??????
6
+�)
= 50(
??????
6
6
+�)
=
50
1
??????
6
6
+�
=
50
4
??????
6
+�

5-∫9??????
8
�??????=9∫??????
8
�??????
= 9(
1
9
??????
9
+�)
= 9(
??????
9
9
+�)
=
9
1
??????
9
9
+�
=
9
9
??????
9
+�

12. Formula 5 ∫??????
??????
�??????=
??????
??????+1
??????+1
+�
1-∫(??????+1)(3??????−2)�?????? = ∫??????(3??????−2)+1(3??????−2)�??????
= ∫??????(3??????)+??????∗−2+1(3??????−2)�??????
= ∫??????(3??????)+??????∗−2+(1(3??????)+1−2)�??????
= 3??????
2
−2??????+(3??????−2)�??????
= ∫3??????
2
+(3??????−2)�??????
= ∫3??????
2
+??????−2�??????
= ∫3??????
2
�??????+∫xdx ∫−2dx
= 3∫??????
2
�??????+∫xdx ∫−2dx
= 3(
1
3
??????
3
+�)+(xdx)+∫−2dx
= 3(
??????
3
3
+�)+(xdx)+∫−2dx
= 3(
??????
3
3
+�)+(
1
2
dx)+∫−2dx
= 3(
??????
3
3
+�)+
??????
2
2
+c+(−2x+c)
= ??????
3
+
1
2
??????
2
−2??????+�

13. 2-∫(??????
5
−2??????)
2
(??????−1)�?????? =
1
5
(??????
5
−2??????)
2
(??????−1)�??????
=
1
5
(??????
5
−2??????)
3
+�
=
1
5
(??????
5
−2??????)
3
3
+�
=
(??????
5
−2??????)
3
15
+�


3-∫(??????
6
−2??????)
9
(??????+2)�??????=
1
6
(??????
6
−2??????)
9
(??????+2??????)�??????
=
1
6
(??????
6
−2??????)
9
+�
=
(??????
6
−2??????)
9
6
+�


4-∫(??????
2
−1??????)(??????−1??????)�?????? =
1
1
(??????
2
−1??????)(??????−1??????)�??????
=
1
1
(??????
2
−1??????)
1
+�
=
??????
2
−1??????
1
+�


5-∫(3??????
5
−2??????)
3
(??????+1)�??????=
3
1
(3??????
5
−2??????)
3
(??????+1)�??????
=
3
1
(3??????
5
−2??????)
3
+�
=
(3??????
5
−2??????)
1
+�