![Integration:
Integration is the reverse process of differentiation. Suppose 𝑑
𝑑𝑥
𝑓(𝑥) = 𝐹(𝑥),
then 𝐹(𝑥) is called ant derivative or integration of the function 𝐹(𝑥) with respect
to x.
In symbol, we have ∫ 𝐹(𝑥)𝑑𝑥 = 𝑓(𝑥) + 𝑐
Again, if 𝑑
𝑑𝑥
[𝐹(𝑥) + 𝑐] 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝐹(𝑥)
Formulae
1. ∫ 𝒌𝒇( 𝒙) 𝒅𝒙 = 𝒌 ∫ 𝒇( 𝒙) 𝒅𝒙 + 𝒄
2. ∫[ 𝒇( 𝒙) + 𝒈( 𝒙)] 𝒅𝒙 = ∫ 𝒇( 𝒙) 𝒅𝒙 + ∫ 𝒈( 𝒙) 𝒅𝒙 + 𝒄
3. ∫ 𝒙 𝒏
𝒅𝒙 =
𝒙 𝒏+𝟏
𝒏+𝟏
+ 𝒄, 𝒏 ≠ −𝟏
4. ∫ 𝒅𝒙 = 𝒙 + 𝒄
5. ∫
𝟏
𝒙
𝒅𝒙 = 𝐥𝐧 𝒙 + 𝒄
6. ∫ 𝒆 𝒙
𝒅𝒙 = 𝒆 𝒙
+ 𝒄
7. ∫ 𝒆 𝒂𝒙
𝒅𝒙 =
𝒆 𝒂𝒙
𝒂
+ 𝒄
8. ∫( 𝒂𝒙 + 𝒃) 𝒏
𝒅𝒙 =
(𝒂𝒙+𝒃) 𝒏+𝟏
𝒂(𝒏+𝟏)
+ 𝒄, 𝒏 ≠ −𝟏
9. ∫
𝟏
𝒂𝒙+𝒃
𝒅𝒙 =
𝐥𝐨𝐠(𝒂𝒙+𝒃)
𝒂
+ 𝒄](https://image.slidesharecdn.com/integration-150302025451-conversion-gate01/85/integration-1-320.jpg)
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Integration
- 1. Integration: Integration is the reverse process of differentiation. Suppose 𝑑 𝑑𝑥 𝑓(𝑥) = 𝐹(𝑥), then 𝐹(𝑥) is called ant derivative or integration of the function 𝐹(𝑥) with respect to x. In symbol, we have ∫ 𝐹(𝑥)𝑑𝑥 = 𝑓(𝑥) + 𝑐 Again, if 𝑑 𝑑𝑥 [𝐹(𝑥) + 𝑐] 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝐹(𝑥) Formulae 1. ∫ 𝒌𝒇( 𝒙) 𝒅𝒙 = 𝒌 ∫ 𝒇( 𝒙) 𝒅𝒙 + 𝒄 2. ∫[ 𝒇( 𝒙) + 𝒈( 𝒙)] 𝒅𝒙 = ∫ 𝒇( 𝒙) 𝒅𝒙 + ∫ 𝒈( 𝒙) 𝒅𝒙 + 𝒄 3. ∫ 𝒙 𝒏 𝒅𝒙 = 𝒙 𝒏+𝟏 𝒏+𝟏 + 𝒄, 𝒏 ≠ −𝟏 4. ∫ 𝒅𝒙 = 𝒙 + 𝒄 5. ∫ 𝟏 𝒙 𝒅𝒙 = 𝐥𝐧 𝒙 + 𝒄 6. ∫ 𝒆 𝒙 𝒅𝒙 = 𝒆 𝒙 + 𝒄 7. ∫ 𝒆 𝒂𝒙 𝒅𝒙 = 𝒆 𝒂𝒙 𝒂 + 𝒄 8. ∫( 𝒂𝒙 + 𝒃) 𝒏 𝒅𝒙 = (𝒂𝒙+𝒃) 𝒏+𝟏 𝒂(𝒏+𝟏) + 𝒄, 𝒏 ≠ −𝟏 9. ∫ 𝟏 𝒂𝒙+𝒃 𝒅𝒙 = 𝐥𝐨𝐠(𝒂𝒙+𝒃) 𝒂 + 𝒄
- 2. 10. ∫ 𝒖𝒗 𝒅𝒙 = 𝒖 ∫ 𝒗 𝒅𝒙 − ∫ { 𝒅𝒖 𝒅𝒙 ∫ 𝒗𝒅𝒙} 𝒅𝒙 11. ∫ 𝒂 𝒙 𝒅𝒙 = 𝒂 𝒙 𝐥𝐧 𝒂 + 𝒄, 𝒂 > 𝟎 12. ∫ 𝐬𝐢𝐧 𝒂𝒙𝒅𝒙 = 𝐜𝐨𝐬 𝒂𝒙 𝒂 + 𝒄 13. ∫ 𝐜𝐨𝐬 𝒂𝒙𝒅𝒙 = 𝐬𝐢𝐧 𝒂𝒙 𝒂 + 𝒄 14. ∫ 𝐬𝐢𝐧 𝒙𝒅𝒙 = − 𝐜𝐨𝐬 𝒙 + 𝒄 15. ∫ 𝐜𝐨𝐬 𝒙𝒅𝒙 = 𝐬𝐢𝐧 𝒙 + 𝒄 16. ∫ 𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙𝒅𝒙 = − 𝐜𝐨𝐭 𝒙 + 𝒄 17. ∫ 𝒔𝒆𝒄 𝟐 𝒙𝒅𝒙 = 𝐭𝐚𝐧 𝒙 + 𝒄 18. ∫ 𝐭𝐚𝐧 𝒙𝒅𝒙 = 𝐥𝐨𝐠 𝐬𝐞𝐜 𝒙 + 𝒄 19. ∫ 𝒄𝒐𝒔𝒆𝒄𝒙 𝒄𝒐𝒕𝒙𝒅𝒙 = −𝒄𝒐𝒔𝒆𝒄𝒙 + 𝒄 20. ∫ 𝐜𝐨𝐭 𝒙𝒅𝒙 = 𝐥𝐨𝐠 𝐬𝐢𝐧 𝒙 + 𝒄 21. ∫ 𝒔𝒆𝒄𝒙 𝒕𝒂𝒏𝒙 𝒅𝒙 = 𝒔𝒆𝒄𝒙 + 𝒄