algunos ejemplos de integrales indefinidas

1. ∫ 𝒙 𝒏
𝒅𝒙 =
𝒙 𝒏+𝟏
𝒏+𝟏
+ 𝑪
∫ 𝟓𝒙 𝟒
𝒅𝒙 = 5∫ 𝑥4
𝑑𝑥
=
5 𝑥4+1
4+1
=
5 𝑥5
5
= 𝑥5
+ 𝐶
∫ 𝟑𝒙 𝟒
𝒅𝒙 = 3∫ 𝑥4
𝑑𝑥
=
3 𝑥4+1
4+1
=
3 𝑥5
5
=
3
5
𝑥5
+ 𝐶
∫ 𝟒𝟖𝒙 𝟕
𝒅𝒙 = 48∫ 𝑥7
𝑑𝑥
=
48 𝑥7+1
7+1
=
48𝑥8
8
=
48
8
𝑥8
= 6𝑥8
+ C

2. ∫ ( 𝒅𝒖 + 𝒅𝒗 − 𝒅𝒘) = ∫ 𝒅𝒖 + ∫ 𝒅𝒗 − ∫ 𝒅𝒘
∫ ( 𝟕𝒙 𝟐
+ 𝟑𝒙 − 𝟐𝒙) 𝒅𝒙 = ∫ 7𝑥2
𝑑𝑥 + ∫ 3𝑥𝑑𝑥 − ∫ 2𝑥𝑑𝑥
= 7∫ 𝑥2
𝑑𝑥 + 3∫ 𝑥𝑑𝑥 − 2∫ 𝑥𝑑𝑥
=
7𝑥2+1
2+1
+
3𝑥1+1
1+1
−
2𝑥1+1
1+1
=
7𝑥3
3
+
3𝑥2
2
−
2𝑥2
2
=
7
3
𝑥3
+
3
2
𝑥2
−
2
2
𝑥2
=
7
3
𝑥3
+
3
2
𝑥2
− 𝑥2
+ 𝐶
∫ ( 𝟑𝒙 𝟑
+ 𝟕𝒙 𝟐
− 𝟗𝒙) 𝒅𝒙 = ∫ 3𝑥3
𝑑𝑥 + ∫ 7𝑥2
𝑑𝑥 − ∫ 9𝑥𝑑𝑥
= 3∫ 𝑥3
𝑑𝑥 + 7∫ 𝑥2
𝑑𝑥 − 9∫ 𝑥𝑑𝑥
=
3𝑥3+1
3+1
+
7𝑥2+1
2+1
−
9𝑥1+1
1+1
=
3𝑥4
4
+
7𝑥3
3
−
9𝑥2
2
=
3
4
𝑥4
+
7
3
𝑥3
−
9
2
𝑥2
+ 𝐶
∫ (𝟖𝒙 𝟒
+ 𝟑𝒙 𝟓
− 𝟔𝒙 𝟐
) 𝒅𝒙 = ∫ 8𝑥4
𝑑𝑥 + ∫ 3𝑥5
𝑑𝑥 − ∫ 6𝑥2
𝑑𝑥
= 8∫ 𝑥4
𝑑𝑥 + 3∫ 𝑥5
𝑑𝑥 − 6∫ 𝑥2
𝑑𝑥
=
8𝑥4+1
4+1
+
3𝑥5+1
5+1
−
6𝑥2+1
2+1
=
8𝑥5
5
+
3𝑥6
6
−
6𝑥3
3
=
8
5
𝑥5
+
3
6
𝑥6
−
6
3
𝑥3
=
8
5
𝑥5
+
3
6
𝑥6
− 2𝑥3
+ 𝐶

3. ∫ 𝒗 𝒏
𝒅𝒗 =
𝒗 𝒏+𝟏
𝒏+𝟏
∫ √ 𝟐 + 𝟑𝒙 𝒅𝒙 = ∫ (2 + 3𝑥)
1
2 𝑑𝑥
=
1
3
∫ (2 + 3𝑥)
1
2 3𝑑𝑥
𝑣 = 2 + 3𝑥 =
1
3
(2+3𝑥)
1
2
+1
1
2
+1
𝑑𝑣 = 3𝑑𝑥 =
1
3
(2+3𝑥)
3
2
3
2
=
1
3
3
2
(2 + 3𝑥)
3
2
=
2
9
(2 + 3𝑥)
3
2
=
2
9
√(2 + 3𝑥)3 + 𝐶
∫ √ 𝟖𝒙 − 𝟏𝟐
𝟑
𝒅𝒙 = ∫ (8𝑥 − 12)
1
3 𝑑𝑥
=
1
8
∫ (8𝑥 − 12)
1
3 8𝑑𝑥
𝑣 = 8𝑥 − 12 =
1
8
(8𝑥−12)
1
3
+1
1
3
+1
𝑑𝑣 = 8𝑑𝑥 =
1
8
(8𝑥−12)
4
3
4
3
=
1
8
4
3
(8𝑥 − 12)
4
3
=
3
32
(8𝑥 − 12)
4
3
=
3
32
√(8𝑥 − 12)43
+ 𝐶

4. ∫ 𝒙 (𝟑𝒙 𝟐
+ 𝟓) 𝟑
𝒅𝒙 = ∫ (3𝑥2
+ 5)3
𝑑𝑥
=
1
6
∫ (3𝑥2
+ 5)3
6𝑑𝑥
𝑣 = 3𝑥2
+ 5 =
1
6
(3𝑥2+5)3+1
3+1
𝑑𝑣 = 6𝑥𝑑𝑥 =
1
6
(3𝑥2+5)4
4
=
1
6
4
1
(3𝑥2
+ 5)4
=
1
24
(3𝑥2
+ 5)4
+ 𝐶
∫ 𝟒𝒙 √𝟖 − 𝟐𝒙 𝟐 𝒅𝒙 = 4∫ 𝑥 √8 − 2𝑥2 𝑑𝑥
= 4∫ 𝑥(8 − 2𝑥2)
1
2 𝑑𝑥
= 4∫ (8− 2𝑥2)
1
2 𝑑𝑥
= 4(−
1
4
(8 − 2𝑥2)
1
2 − 4𝑥𝑑𝑥
𝑣 = 8 − 2𝑥2
= 4 (−
1
4
(8−2𝑥2)
1
2
+1
1
2
+1
)
𝑑𝑣 = −4𝑥𝑑𝑥 = 4(−
1
4
(8−2𝑥2)
3
2
3
2
)
=
−
4
4
(8−2𝑥2)
3
2
3
2
=
−
4
4
3
2
(8 − 2𝑥2)
3
2
= −
8
12
(8 − 2𝑥2)
3
2
= −
2
3
(8 − 2𝑥2)
3
2
= −
2
3
√(8 − 2𝑥2)3

5. ∫ 𝒆 𝒗
𝒅𝒗 = 𝒆 𝒗
+ 𝑪
∫ 𝒆 𝟒𝒙
𝒅𝒙 =
1
4
∫ 𝑒4𝑥
4𝑑𝑥
𝑣 = 4𝑥 =
1
4
𝑒4𝑥
+ 𝐶
𝑑𝑣 = 4𝑑𝑥
∫ 𝒆 𝟐𝒙 𝟐
𝒅𝒙 =
1
4
∫ 𝑒2𝑥2
4𝑥𝑑𝑥
𝑣 = 2𝑥2
=
1
4
𝑒2𝑥2
+ 𝐶
𝑑𝑣 = 4𝑥𝑑𝑥
∫
𝒆 𝒙−𝟒
𝒅𝒙
𝒙 𝟓
= −
1
4
∫ 𝑒 𝑥−4
− 4𝑥−3
𝑑𝑥
𝑣 = 𝑥−4
= −
1
4
𝑒 𝑥−4
+ 𝐶
𝑑𝑣 = −4𝑥−3
𝑑𝑥
Integral de forma ∫ 𝒕𝒂𝒏 𝒏
𝒗 𝒅𝒗 con n par o impar
Identidad trigonométrica 𝑡𝑎𝑛2
𝑥 = 𝑠𝑒𝑐2
𝑥 − 1
∫ 𝒕𝒂𝒏 𝟑
𝒙 𝒅𝒙 = ∫ 𝑡𝑎𝑛x 𝑡𝑎𝑛2
𝑥 𝑑𝑥
= ∫ tan x 𝑡𝑎𝑛2
𝑥 𝑑𝑥
𝑣 = 𝑡𝑎𝑛𝑥 = ∫ 𝑡𝑎𝑛𝑥 ∗ 𝑠𝑒𝑐2
𝑥𝑑𝑥 − ∫ 𝑡𝑎𝑛𝑥 𝑑𝑥
𝑑𝑣 = 𝑠𝑒𝑐2
𝑥𝑑𝑥 = ∫ 𝑣𝑑𝑣 − ∫ 𝑡𝑎𝑛𝑥 𝑑𝑥
=
𝑣2
2
− (−𝑙𝑛 ↾ 𝑐𝑜𝑠𝑥 ↿ +𝐶
=
1
2
𝑡𝑎𝑛2
𝑥 + ↾ 𝑐𝑜𝑠𝑥 ↿ +𝐶

6. ∫ 𝒙 √ 𝒙 − 𝟕 𝒅𝒙 =
𝑤 = 𝑥 − 7
𝑥 = 𝑤 + 7
𝑑𝑥
𝑑𝑤
= 1 = ∫ ( 𝑤 + 7) √ 𝑤 + 7 − 7 𝑑𝑤
𝑑𝑥 = 𝑑𝑤 = ∫ ( 𝑤 + 7) √ 𝑤 𝑑𝑤
= ∫ ( 𝑤 + 7) 𝑤
1
2 𝑑𝑤
= ∫ (𝑤
3
2 + 7𝑤
1
2) 𝑑𝑤
= ∫ 𝑤
3
2 𝑑𝑤 + 7∫ 𝑤
1
2 𝑑𝑤
=
𝑤
3
2
+1
3
2
+1
+
7𝑤
1
2
+1
1
2
+1
=
𝑤
5
2
5
2
+
7𝑤
3
2
3
2
=
5
2
𝑤
5
2 +
14
3
𝑤
3
2
=
5
2
( 𝑥 − 7)
5
2 +
14
3
( 𝑥 − 7)
3
2
=
5
2
( 𝑥 − 7)
4
2( 𝑥 − 7)
1
2 +
14
3
( 𝑥 − 7)
2
2 ( 𝑥 − 7)
1
2
=
5
2
( 𝑥 − 7)2
√ 𝑥 − 7 +
14
3
( 𝑥 − 7)√ 𝑥 − 7 + 𝐶

7. ∫ 𝒙 √ 𝒙 − 𝟕 𝒅𝒙 =
𝑤 = √ 𝑥 − 7
𝑥 = 𝑤2
+ 7
𝑑𝑥
𝑑𝑤
= 2𝑤 = ∫ ( 𝑤2
+ 7) 𝑤 2𝑤𝑑𝑤
𝑑𝑥 = 2𝑤𝑑𝑤 = ∫ ( 𝑤2
+ 7) 2𝑤2
𝑑𝑤
= ∫ (2𝑤4
+ 14𝑤2) 𝑑𝑤
= 2∫ 𝑤4
𝑑𝑤 + 14∫ 𝑤2
𝑑𝑤
=
2𝑤4+1
4+1
+
14𝑤2+1
2+1
=
2𝑤5
5
+
14𝑤3
3
=
2
5
𝑤5
+
14
3
𝑤3
=
2
5
(√ 𝑥 − 7)
5
+
14
3
(√ 𝑥 − 7)
3
=
2
5
(√ 𝑥 − 7)
4
√ 𝑥 − 7 +
14
3
(√ 𝑥 − 7)
2
√ 𝑥 − 7 + 𝐶

8. ∫
𝟑𝒙−𝟒
𝒙 𝟐+𝟒𝒙
3𝑥−4
𝑥2+4𝑥
=
𝐴
𝑥
+
𝐵
𝑥+4
3𝑥−4
𝑥2+4𝑥
=
𝐴( 𝑥+4)+𝐵𝑥
𝑥( 𝑥+4)
3𝑥−4
𝑥2+4𝑥
=
𝐴𝑥+4𝐴+𝐵𝑥
𝑥( 𝑥+4)
3𝑥 − 4 = ( 𝐴 + 𝐵) 𝑥 + 4𝐴
𝐴 + 𝐵 = 3 4 + 𝐵 = 3
𝐴 = 1 𝐵 = 3 − 4
𝐵 = −1
∫
𝟑𝒙−𝟒
𝒙 𝟐+𝟒𝒙
𝒅𝒙 = ∫ (
4
𝑥
+
−1
𝑥+4
) 𝑑𝑥
= 4∫
𝑑𝑥
𝑥
− 1∫
𝑑𝑥
𝑥+4
= 4𝑙𝑛 ↾ 𝑥 ↿ −1𝑙𝑛 ↾ 𝑥 + 4 ↿ +𝑙𝑛𝐶
= 𝑙𝑛 ↾ 𝑥 ↿4
𝑙𝑛 ↾ 𝑥 + 4 ↿−1
+ 𝑙𝑛𝐶
= 𝑙𝑛
𝐶 ↾ 𝑥 ↿4
↾ 𝑥 + 4 ↿1

9. ∫
𝟐𝒙 − 𝟖
𝒙 𝟐 − 𝟒
2𝑥−8
𝑥2−4
=
𝐴
𝑥−2
+
𝐵
𝑥+2
2𝑥−8
𝑥2−4
=
𝐴( 𝑥+2)+𝐵(𝑥−2)
(𝑥−2)( 𝑥+2)
2𝑥−8
𝑥2−4
=
𝐴𝑥+2𝐴+𝐵𝑥−2𝐵
(𝑥−2)( 𝑥+2)
2𝑥 − 8 = ( 𝐴 + 𝐵) 𝑥 + 2𝐴 − 2𝐵
(2) 𝐴 + 𝐵 = 2
2𝐴 − 2𝐵 = −8
2𝐴 − 2𝐵 = −8
2𝐴 + 2𝐵 = 4 2(−1) − 2𝐵 = −8
2𝐴 − 2𝐵 = −8 −2 − 2𝐵 = −8
4𝐴 = −4 −2𝐵 = −8 + 2
𝐴 =
−4
4
−2𝐵 = −6
𝐴 = −1 𝐵 =
−6
−2
B = 3
∫
𝟐𝒙−𝟖
𝒙 𝟐−𝟒
𝒅𝒙 = ∫ (
−1
𝑥−2
−
3
𝑥+2
) 𝑑𝑥
= −1∫
𝑑𝑥
𝑥−2
− 3∫
𝑑𝑥
𝑥+2
= −1𝑙𝑛 ↾ 𝑥 − 2 ↿ −3𝑙𝑛 ↾ 𝑥 + 2 ↿ +𝑙𝑛𝐶
= 𝑙𝑛 ↾ 𝑥 − 2 ↿−1
𝑙𝑛 ↾ 𝑥 + 2 ↿−3
+ 𝑙𝑛𝐶
= 𝑙𝑛
𝐶
↾ 𝑥 − 2 ↿1 ↾ 𝑥 + 2 ↿3

10. ∫ 𝒖𝒅𝒗 = 𝒖𝒗 − ∫ 𝒗𝒅𝒖
∫ 𝒙 𝟐
𝒆 𝒙
𝒅𝒙
𝑢 = 𝑥2
𝑑𝑢 = 2𝑥𝑑𝑥
𝑑𝑣 = 𝑒 𝑥
𝑑𝑥 = 𝑥2
𝑒 𝑥
− ∫ 𝑒 𝑥
2𝑥𝑑𝑥
𝑣 = 𝑒 𝑥
+ 𝐶 = 𝑥2
𝑒 𝑥
− 2∫ 𝑒 𝑥
𝑥𝑑𝑥
𝑢 = 𝑥
𝑑𝑢 = 𝑑𝑥
𝑑𝑣 = 𝑒 𝑥
𝑑𝑥 = 𝑥2
𝑒 𝑥
− 2[ 𝑥𝑒 𝑥
− ∫ 𝑒 𝑥
𝑑𝑥]
𝑣 = 𝑒 𝑥
+ 𝐶 = 𝑥2
𝑒 𝑥
− 2( 𝑥𝑒 𝑥
− 𝑒 𝑥
𝑑𝑥)
= 𝑥2
𝑒 𝑥
− 2𝑥𝑒 𝑥
+ 2𝑒 𝑥
+ 𝐶
∫ 𝟐𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙
𝑢 = 2𝑥
𝑑𝑢 = 2𝑑𝑥
𝑑𝑣 = 𝑐𝑜𝑠𝑥 𝑑𝑥 = 2𝑥 𝑠𝑒𝑛𝑥 − ∫ 𝑠𝑒𝑛𝑥 2𝑑𝑥
𝑣 = 𝑠𝑒𝑛𝑥 + 𝐶 = 2𝑥 𝑠𝑒𝑛𝑥 − 2∫ 𝑠𝑒𝑛𝑥 𝑑𝑥
= 2𝑥 𝑠𝑒𝑛𝑥 − 2( 𝑐𝑜𝑠𝑥)
= 2𝑥 𝑠𝑒𝑛𝑥 + 2 𝑐𝑜𝑠𝑥 + 𝐶