A brief introduction to the concept of the integral of a function as the inverse of the derivative function.

1. Applied Calculus I – MTH 271
Integrals of Functions (Anti-Derivatives)
Functions and their Inverses
Examples:
Function Inverse
𝑦 = 3𝑥 − 7
𝑦 =
𝑥 + 7
3

2. 1
3
7
-1
2
14
𝑦 = 3𝑥 − 7
X Y
- 1
2
14
1
3
7
𝑦 =
𝑥 + 7
3
X Y
Function:
Inverse Function (of 𝑦 = 3𝑥 − 7

3. Applied Calculus I – MTH 271
Integrals of Functions (Anti-Derivatives)
Functions and their Inverses
Examples:
Function Inverse
𝑦 = 3𝑥 − 7
𝑦 =
𝑥 + 7
3
𝑦 = 𝑒 𝑥
𝑦 = ln 𝑥
𝑦′
𝑦
𝑑𝑦
𝑑𝑥

4. 𝑦 = 5𝑥4
Function Derivative
𝑦′ = 20𝑥3𝑦 = 5𝑥4 + 35
𝑦 = 5𝑥4 − 163
The Derivative Function
This function (the derivative function) is not one-to-one so its
inverse is not a function.
The inverse (the integral of 𝑦′ = 20𝑥) can be expressed as:
𝑦 = 5𝑥4 + 𝐶 , C being any real number.

5. 𝑓(𝑥) = 5𝑥4
+ 𝐶
𝑓(𝑥) = 20𝑥3If
represents a mathematical model of some process,
then we need more information to identify its integral function.
Example, if we know that at x = 0, y = 25, then C = 25,
𝑓 𝑥 = 5𝑥4 + 25
then,