Calculus

1. Integration: The Basic Exponential and
Logarithmic Functions
ENGR. ROGELIO RUZCKO TOBIAS

2. Integration: The Basic Exponential Form
There are two basic formulas for the integration of exponential
functions.
1. ‫׬‬ 𝑎𝑢
𝑑𝑢 =
𝑎𝑢
ln |𝑎|
+ 𝐶, 𝑎 > 0, ≠ 1
2. ‫׬‬ 𝑒𝑢
𝑑𝑢 = 𝑒𝑢
+ 𝐶

3. Examples
Evaluate the following functions:
1. ‫׬‬ 3𝑒𝑥 𝑑x
2. ‫׬‬(2𝑥3 + 𝑒𝑥)𝑑𝑥
3. ‫׬‬ 𝑥𝑒𝑥2
𝑑𝑥
4. ‫׬‬
𝑒 𝑥
𝑥
𝑑𝑥
5. ‫׬‬ 𝑒𝑥(𝑒𝑥 + 1)2 𝑑𝑥

4. Integration: The Basic Logarithmic Form
න
𝑑𝑢
𝑢
= 𝑙𝑛 𝑢 + 𝐶

5. Examples
Evaluate the following functions:
1. ‫׬‬
2
𝑥
𝑑𝑥
2. ‫׬‬
1
4𝑥−1
𝑑𝑥
3. ‫׬‬
𝑒𝑥
1+𝑒𝑥 𝑑𝑥
4. ‫׬‬
𝑥+1
𝑥2+2𝑥
𝑑𝑥
5. ‫׬‬
𝑒𝑥+𝑒−𝑥
𝑒𝑥−𝑒−𝑥 𝑑𝑥

6. Seatwork
Evaluate the following functions:
1. ‫׬‬ 3𝑥
𝑑𝑥
2. ‫׬‬(2𝑥3
+ 5𝑥
)𝑑𝑥
3. ‫׬‬ 7𝑥2
𝑥𝑑𝑥
4. ‫׬‬
3𝑥+4𝑥
5𝑥 𝑑𝑥
5. ‫׬‬ 4𝑥
(4𝑥
+ 1)2
6. ‫׬‬
1
𝑥𝑙𝑛𝑥
𝑑𝑥
7. ‫׬‬
1
𝑥(3+𝑙𝑛𝑥)
𝑑𝑥
8. ‫׬‬
2𝑥3
𝑥4+1
𝑑𝑥
9. ‫׬‬
𝑥2+1
𝑥3+3𝑥
𝑑𝑥
10. ‫׬‬
𝑥−3
𝑥−2+4
𝑑𝑥

7. References:
Web Pages:
Mathalino.com
Mathworld.wolfram.com