Lecture #21: Antiderivatives & Differential Equations
MATH 145 – Calculus I
November 6, 2025
Inverse Operations
In a previous lecture, we talked about inverse functions and inverse operations.
For example, if I add \(4\) to a number, I can subtract \(4\) from the answer to reverse or the addition, as subtraction is the inverse operation to addition.
Or if I take \(e\) to the fourth power, \(e^4\), if I want to reverse that or it, I can take the natural logarithm, which is the inverse operation of exponentiation.
If we think of differentiation as an mapping one function \(f\) to its derivative \(f'\), then what would be the equivalent ?
Antiderivatives
If \(f(x)\) is a function, then an antiderivative of \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\).
Example: Find an antiderivative for \(f(x) = 2x\).
Integral Notation
If \(f(x)\) is a function, then the most general antiderivative of \(f(x)\) is denoted by:
\[ \int f(x)\,dx \]
Example:
\[ \int 2x\,dx = x^2 + C \]
Rules for Finding Antiderivatives
(Inverse) Power Rule:
\[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\textrm{for all $n \ne 1$}. \]
Rules for Finding Antiderivatives
Sum Rule:
\[ \int \left( f(x) + g(x) \right)\,dx = \int f(x)\,dx + \int g(x)\,dx. \]
Rules for Finding Antiderivatives
Constant Multiple Rule:
\[ \int cf(x)\,dx = c \int f(x)\,dx. \]
Example: Antiderivatives
Example: Find \(\displaystyle\int 5x + 9 \, dx\).
Example: Find \(\displaystyle\int 2x^2 - 3x + 7 \, dx\).
Example: Find \(\displaystyle\int \dfrac{(x+1)(x-2)}{x^4} \, dx\).
Example: If \(F'(x) = 6x^3\), find \(F(x)\).
Example: If \(F'(x) = 6x^3\), find \(F(x)\) such that \(F(1) = 3\).
Differential Equations and Initial Value Problems
Example: A car is traveling at the rate of \(88\,\frac{ft}{sec}\) when the brakes are applied. The car begins decelerating at a constant rate of \(15\,\frac{ft}{sec^2}\).
- How many seconds before the car stops?
- How far does the car travel during that time?
Differential Equations and Initial Value Problems
Example: The velocity function is \(v(t) = t^2 - 3t + 2\) for a particle moving along a line. Find the displacement (net distance covered) of the particle during the time interval \([-2,5]\).
