P3 Integration_Must know!!
Mother
Child
Integrate
=
Integrate
=
𝑥 𝑛+1
𝑛+1
(𝑎𝑥 + 𝑏)𝑛
2. 𝑠𝑖𝑛𝑥
- 𝑐𝑜𝑠𝑥
sin(𝑎𝑥 + 𝑏)
- 𝑐𝑜𝑠(𝑎𝑥 + 𝑏)/𝑎
3. 𝑐𝑜𝑠𝑥
𝑠𝑖𝑛𝑥
𝑐𝑜𝑠(𝑎𝑥 + 𝑏)
𝑠𝑖𝑛(𝑎𝑥 + 𝑏)/𝑎
4. 𝑠𝑒𝑐 2 𝑥
𝑡𝑎𝑛𝑥
𝑠𝑒𝑐 2 (𝑎𝑥 + 𝑏)
𝑡𝑎𝑛(𝑎𝑥 + 𝑏)/𝑎
𝑐𝑜𝑠𝑒𝑐 2 (𝑎𝑥 + 𝑏)
−𝑐𝑜𝑡(𝑎𝑥 + 𝑏)/𝑎
1. 𝑥 𝑛
5. 𝑐𝑜𝑠𝑒𝑐 2 𝑥
−𝑐𝑜𝑡𝑥
(𝑎𝑥+𝑏)𝑛+1
(𝑛+1)𝑎
6. 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥
𝑠𝑒𝑐𝑥
𝑠𝑒𝑐(𝑎𝑥 + 𝑏)𝑡𝑎𝑛(𝑎𝑥 + 𝑏)
𝑠𝑒𝑐(𝑎𝑥 + 𝑏)/𝑎
7. 𝑐𝑜𝑠𝑒𝑐𝑥𝑐𝑜𝑡𝑥
−𝑐𝑜𝑠𝑒𝑐𝑥
𝑐𝑜𝑠𝑒𝑐(𝑎𝑥 + 𝑏)𝑐𝑜𝑡(𝑎𝑥 + 𝑏)
−𝑐𝑜𝑠𝑒𝑐(𝑎𝑥 + 𝑏)/𝑎
8. 𝑒 𝑥
𝑒𝑥
𝑒 (𝑎𝑥+𝑏)
𝑒 (𝑎𝑥+𝑏) /𝑎
𝑚(𝑎𝑥+𝑏)
𝑚(𝑎𝑥+𝑏)
𝑎 ln 𝑚
𝑚𝑥
ln 𝑚
9. 𝑚 𝑥
10.



1
𝑥
1
(𝑎𝑥+𝑏)
𝑙𝑛|𝑥|
𝑙𝑛|𝑎𝑥 + 𝑏|/𝑎
The Child column is used when the x in Mother is replaced by a linear function of x, ax + b
The 10 lines in the left column (Mother) are the “reverse” of the 10 lines from Differentiation file.
Therefore, if you memorize the differentiation part, the integration on the left column is not
difficult.
the Child’s integral is “automatic”. The pattern is simple to understand; just need to divide by 𝑎, the
derivative of ax + b.
Integration by Recognition (Standard Pattern)
Pattern 1
Pattern 2
∫ 𝑓 ′ (𝑥)[𝑓(𝑥)]𝑘 𝑑𝑥 =
[𝑓(𝑥)]𝑘+1
𝑘+1
∫
𝑓′ (𝑥)
𝑓(𝑥)
𝑑𝑥 = 𝑙𝑛|𝑓(𝑥)|
Pattern 3
∫ 𝑓 ′ (𝑥)𝑒 𝑓(𝑥) 𝑑𝑥 = 𝑒 𝑓(𝑥)
** In each case, you’ll see that when you differentiate RHS, you will get the integrand **
Example
∫ 6𝑥(3𝑥 2 + 1)4 𝑑𝑥 =
(3𝑥 2 +1)5
5
3𝑥 2
∫ 𝑥 3 +1 𝑑𝑥 = ln|𝑥 3 + 1|
* linked to Pattern 1 since 6𝑥 is the derivative of 3𝑥 2 + 1*
* linked to Pattern 2 since 3𝑥 2 (numerator) is the derivative of 𝑥 3 + 1
(denominator) *
∫(𝑐𝑜𝑠𝑥)𝑒 𝑠𝑖𝑛𝑥 𝑑𝑥 = 𝑒 𝑠𝑖𝑛𝑥
* linked to Pattern 3 since cos x is the derivative of sin x*