Table 1: Basic Integral Forms
1. ∫ 𝑑𝑢 = 𝑢 + 𝐶
10. ∫ tan 𝑢 𝑑𝑢 = 𝑙𝑛 |sec 𝑢| + 𝐶
2. ∫ 𝑎 𝑑𝑢 = 𝑎𝑢 + 𝐶
11. ∫ cot 𝑢 𝑑𝑢 = 𝑙𝑛 |sin 𝑢| + 𝐶
3. ∫ 𝑢𝑛 𝑑𝑢 =
𝑢𝑛+1
12. ∫ sec 𝑢 𝑑𝑢 = 𝑙𝑛 |sec 𝑢 − tan 𝑢| + 𝐶
+ 𝐶 , 𝑛 ≠ −1
𝑛+1
4. ∫ sin 𝑢 𝑑𝑢 = −cos 𝑢 + 𝐶
13. ∫ csc 𝑢 𝑑𝑢 = 𝑙𝑛 |csc 𝑢 − cot 𝑢| + 𝐶
5. ∫ cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶
14. ∫ 𝑎𝑢 𝑑𝑢 = 𝑙𝑛 𝑢 + 𝐶
6. ∫ sec 2 𝑢 𝑑𝑢 = tan 𝑢 + 𝐶
15. ∫
7. ∫ csc 2 𝑢 𝑑𝑢 = −cot 𝑢 + 𝐶
16. ∫
8. ∫ sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢 + 𝐶
17. ∫
9. ∫ csc 𝑢 cot 𝑢 𝑑𝑢 = −csc 𝑢 + 𝐶
18. ∫
𝑎𝑢
20. ∫
21. ∫
22. ∫
𝑑𝑢
1
𝑎 + 𝑏𝑢
𝑢
𝑑𝑢 = 𝑏2 (𝑎 + 𝑏𝑢 − 𝑙𝑛 |𝑎 + 𝑏𝑢|) + 𝐶
1
1
𝑢(𝑎 + 𝑏𝑢)
𝑢2
𝑎 + 𝑏𝑢
𝑢
1
1
1 + 𝑢2
2
𝑢√(𝑢2 − 1)
27. ∫
24. ∫
25. ∫
𝑢
(𝑎 + 𝑏𝑢)2
1
𝑏
𝑎 + 𝑏𝑢
= − 𝑎𝑢 + 𝑎2 𝑙𝑛 |
1
𝑢
29. ∫
|+𝐶
30. ∫
𝑎
= 𝑏2 (𝑎+𝑏𝑢 𝑙𝑛|𝑎 + 𝑏𝑢|) + 𝐶
𝑑𝑢
1
1
31. ∫
𝑢
= 𝑎(𝑎+𝑏𝑢) + 𝑎2 𝑙𝑛 |𝑎 + 𝑏𝑢| + 𝐶
𝑢(𝑎+𝑏𝑢)2
32. ∫
Table 5: Integrands Involving √𝒂𝟐 − 𝒖𝟐 , a > 0
33. ∫
𝑑𝑢
√𝑎2 − 𝑢2
36. ∫
𝑑𝑢
𝑢√𝑎2 − 𝑢2
1
𝑢
𝑎2
2
2
𝑎 + √𝑎2 − 𝑢2
= 𝑎 𝑙𝑛 |
𝑑𝑢
𝑢2 √𝑎2 − 𝑢2
𝑑𝑢
𝑎 2 + 𝑢2
𝑑𝑢
𝑎 2 − 𝑢2
𝑑𝑢
𝑢2 − 𝑎 2
1
𝑢
= 𝑎 Arc tan
𝑎
1
𝑢+𝑎
1
𝑢−𝑎
+𝐶
= 2𝑎 𝑙𝑛 |𝑢−𝑎| + 𝐶
= 2𝑎 𝑙𝑛 |𝑢+𝑎| + 𝐶
1
𝑢
𝑑𝑢
√𝑢2 ± 𝑎2
= 𝑙𝑛 |𝑢 + √𝑢2 ± 𝑎2 | + 𝐶
𝑑𝑢
𝑢√𝑢2 − 𝑎2
𝑑𝑢
𝑢√𝑢2 + 𝑎2
𝑑𝑢
𝑢2 √𝑢2 ± 𝑎2
1
𝑢
1
𝑎 + √𝑢2 + 𝑎2
= 𝑎 Arc sec 𝑎 + 𝐶
= 𝑎 𝑙𝑛 |
=±
𝑢
√𝑢2 ± 𝑎2
𝑎2 𝑢
|+𝐶
+𝐶
Table 6: Integrands Involving Trigonometric
Functions
𝑢
= Arc sin 𝑎 + 𝐶
34. ∫ √𝑎2 − 𝑢2 𝑑𝑢 = √𝑎2 − 𝑢2 +
35. ∫
𝑑𝑢 = Arc sec 𝑢 + 𝐶
Table 4: Integrands Involving √𝒖𝟐 ± 𝒂𝟐 , a > 0
𝑎 𝑙𝑛 |𝑎 + 𝑏𝑢|] + 𝐶
23. ∫
𝑑𝑢 = Arc tan 𝑢 + 𝐶
1
𝑑𝑢 = 𝑏2 [2 (𝑎 + 𝑏𝑢)2 − 2𝑎(𝑎 + 𝑏𝑢) +
𝑑𝑢
2
𝑢 (𝑎 + 𝑏𝑢)
𝑑𝑢 = Arc sin 𝑢 + 𝐶
1
28. ∫
𝑑𝑢 = 𝑎 𝑙𝑛 |𝑎 + 𝑏𝑢| + 𝐶
1
= 𝑙𝑛 |𝑢| + 𝐶
√(1 − 𝑢2 )
26. ∫
= 𝑏 𝑙𝑛 |𝑎 + 𝑏𝑢| + 𝐶
1
𝑎 + 𝑏𝑢
𝑢
Table 3: Integrands Involving a2 + u2, a > 0
Table 2: Integrands Involving a + bu
19. ∫
𝑑𝑢
|+𝐶
= − 𝑎2𝑢 √𝑎2 − 𝑢2 + 𝐶
𝑢
Arc sin 𝑎 + 𝐶
1
1
1
1
37. ∫ sin2 𝑢 𝑑𝑢 = 2 𝑢 − 4 𝑠𝑖𝑛 2𝑢 + 𝐶
38. ∫ cos2 𝑢 𝑑𝑢 = 2 𝑢 + 4 𝑠𝑖𝑛 2𝑢 + 𝐶
39. ∫ tan2 𝑢 𝑑𝑢 = tan 𝑢 − 𝑢 + 𝐶
40. ∫ cot 2 𝑢 𝑑𝑢 = −cot 𝑢 − 𝑢 + 𝐶
42. ∫ 𝑙𝑛 𝑢 𝑑𝑢 = 𝑢 𝑙𝑛 𝑢 − 𝑢 + 𝐶
Table 1: Derivatives of Polynomial
1. 𝐷𝑥 (𝑐) = 0
4. 𝐷𝑥 [(𝑓(𝑥) + 𝑔(𝑥)] = (𝑓′(𝑥) + 𝑔′(𝑥)
2. 𝐷𝑥 (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1
5. 𝐷𝑥 [(𝑓(𝑥)𝑔(𝑥)] = (𝑓(𝑥)𝑔′(𝑥) + 𝑓′(𝑥)𝑔(𝑥)
3. 𝐷𝑥 (𝑐𝑓(𝑥)) = 𝑐𝑓′(𝑥)
6. 𝐷𝑥 [𝑔(𝑥)] =
𝑓(𝑥)
Table 2: Derivatives of Trigonometric and Exponential Functions
1. 𝐷𝑥 (sin 𝑥) = cos 𝑥
2. 𝐷𝑥 (cos 𝑥) = −sin 𝑥
3. 𝐷𝑥 (tan 𝑥) = sec 2 𝑥
4. 𝐷𝑥 (sec 𝑥) = sec 𝑥 tan 𝑥
6. 𝐷𝑥 (cot 𝑥) = −csc 2 𝑥
7. 𝐷𝑥 (csc 𝑥) = −csc 𝑥 tan 𝑥
1. 𝐷𝑥 (𝑒 𝑥 ) = 𝑒 𝑥
𝑔(𝑥)𝑓 ′(𝑥) −𝑓(𝑥)𝑔′(𝑥)
[𝑔(𝑥)]2