Section 5.2, Day 1- Day 2 Integration

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1 Section 5.2, 5.4-5.7 Day 1- Day 2 Integration
AP Calculus AB

2 Learning Targets Integrate logarithmic functions
Integrate exponential functions Integrate inverse trigonometric functions Apply integration by substitution to integration the above functions

3 Integration of Logarithmic Functions
1 𝑢 𝑑𝑢 = ln 𝑢 +𝐶

4 Level 1: Example 1 Find 𝑥 𝑑𝑥 𝑥 𝑑𝑥 =2 ln 𝑥 +𝐶

5 Level 1: Example 2 Find 4 4𝑥−1 𝑑𝑥 1 𝑢 , 𝑢=4𝑥−1, 𝑑𝑢=4𝑑𝑥,
1 𝑢 𝑑𝑢 = ln 𝑢 +𝑐= ln 4𝑥−1 +𝐶

6 Level 2: Example 1 Find 6 𝑥 2 +2 𝑥 3 +𝑥 𝑑𝑥
1 𝑢 , 𝑢= 𝑥 3 +𝑥, 𝑑𝑢=3 𝑥 2 +1𝑑𝑥 2 𝑑𝑢 𝑢 =2 ln 𝑢 +𝐶=2 ln 𝑥 3 +𝑥 +𝐶

7 Level 2: Example 2 Find 1 3𝑥+2 𝑑𝑥 1 𝑢 , 𝑢=3𝑥+2, 𝑑𝑢=3𝑑𝑥, 𝑑𝑥= 𝑑𝑢 3
1 𝑢 𝑑𝑢 3 = 𝑢 𝑑𝑢 = 1 3 ln 𝑢 +𝐶= 1 3 ln 3𝑥+2 +𝐶

8 Level 1: Example 3 Find 𝑒 𝑥 1+ 𝑒 𝑥 𝑑𝑥 1 𝑢 , 𝑢=1+ 𝑒 𝑥 , 𝑑𝑢= 𝑒 𝑥 𝑑𝑥
1 𝑢 𝑑𝑢 = ln 𝑢 +𝐶= ln 1+ 𝑒 𝑥 +𝐶

9 Level 3: Example 1 Find 2𝑥 𝑥+1 2 𝑑𝑥 𝑢=𝑥+1, 𝑑𝑢=𝑑𝑥, 𝑥=𝑢−1
2 𝑢−1 𝑢 2 𝑑𝑢 = 2𝑢−2 𝑢 2 𝑑𝑢 = 2 𝑢 − 2 𝑢 2 𝑑𝑢 = 𝑢 −2 𝑢 −2 𝑑𝑢 2 ln 𝑢 +2 𝑢 −1 +𝐶=2 ln 𝑥 𝑥+1 +𝐶

10 Integration of Trigonometric Functions
tan 𝑢 𝑑𝑢 =− ln cos 𝑢 +𝐶 cot 𝑢 𝑑𝑢 = ln sin 𝑢 +𝐶 sec 𝑢 𝑑𝑢 = ln sec 𝑢 + tan 𝑢 +𝐶 csc 𝑢 𝑑𝑢 =− ln csc 𝑢 + cot 𝑢 +𝐶

11 Integration of Exponential Functions
𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢 +𝐶 And 𝑎 𝑥 𝑑𝑥 = 1 ln 𝑎 𝑎 𝑥 +𝐶

12 Level 1: Example 1 Find 3 𝑒 3𝑥+1 𝑑𝑥 𝑒 𝑢 , 𝑢=3𝑥+1, 𝑑𝑢=3𝑑𝑥
𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢 +𝐶= 𝑒 3𝑥+1 +𝐶

13 Level 1: Example 2 Find −2𝑥 𝑒 − 𝑥 2 𝑑𝑥 𝑒 𝑢 , 𝑢=− 𝑥 2 , 𝑑𝑢=−2𝑥𝑑𝑥
𝑒 𝑢 𝑑𝑢 = 𝑒 − 𝑥 2 +𝐶

14 Level 1: Example 3 Find 𝑥 𝑑𝑥 = 1 ln 𝑥 +𝐶

15 Level 2: Example 1 Find 𝑒 1 𝑥 𝑥 2 𝑑𝑥 𝑒 𝑢 , 𝑢= 1 𝑥 , 𝑑𝑢=− 1 𝑥 2 𝑑𝑥
𝑒 𝑢 , 𝑢= 1 𝑥 , 𝑑𝑢=− 1 𝑥 2 𝑑𝑥 − 𝑒 𝑢 𝑑𝑢 = −𝑒 1 𝑥 +𝐶

16 Level 2: Example 2 Find sin 𝑥 𝑒 cos 𝑥 𝑑𝑥
𝑒 𝑢 , 𝑢= cos 𝑥 , 𝑑𝑢=− sin 𝑥 𝑑𝑥, sin 𝑥 𝑑𝑥=−𝑑𝑢 𝑒 𝑢 (−𝑑𝑢) =− 𝑒 𝑢 𝑑𝑢 =− 𝑒 𝑢 +𝐶=− 𝑒 cos 𝑥 +𝐶

17 Level 2: Example 3 Find 𝑒 −3𝑥 𝑑𝑥 𝑒 𝑢 , 𝑢=−3𝑥, 𝑑𝑢=−3𝑑𝑥, 𝑑𝑥= 𝑑𝑢 −3
𝑒 𝑢 𝑑𝑢 −3 =− 𝑒 𝑢 𝑑𝑢 =− 1 3 𝑒 −3𝑥 +𝐶

18 Integration of Inverse Trigonometric Functions
𝑑𝑢 𝑎 2 − 𝑢 2 = arcsin 𝑢 𝑎 +𝐶 𝑑𝑢 𝑢 𝑢 2 − 𝑎 2 = 1 𝑎 arcsec 𝑢 𝑎 +𝐶 𝑑𝑢 𝑎 2 + 𝑢 2 = 1 𝑎 arctan 𝑢 𝑎 +𝐶

19 Example 1 Find 𝑑𝑥 2+9 𝑥 2 𝑎= 2 , 𝑢=3𝑥, 𝑑𝑢=3𝑑𝑥, 𝑑𝑥= 𝑑𝑢 3
𝑎= 2 , 𝑢=3𝑥, 𝑑𝑢=3𝑑𝑥, 𝑑𝑥= 𝑑𝑢 3 𝑑𝑢 3 𝑎 𝑢 2 = 𝑑𝑢 𝑎 𝑢 2 = arctan 3𝑥 𝐶

20 Example 2 Find 𝑑𝑥 𝑥 2 −4𝑥+7 Complete the square using 𝑏 2 2
𝑑𝑥 𝑥 2 −4𝑥+4 −4+7 = 𝑑𝑥 𝑥− 𝑎= 3 , 𝑢=𝑥−2, 𝑑𝑢=𝑑𝑥 𝑑𝑢 𝑢 2 + 𝑎 2 = arctan 𝑥− 𝐶

21 Example 3 Find 𝑑𝑥 4− 𝑥 2 𝑎=2, 𝑢=𝑥, 𝑑𝑢=𝑑𝑥 𝑑𝑢 𝑎 2 − 𝑢 2 = arcsin 𝑥 2 +𝐶