Welcome back to Professor Baker's Math Class! In Section 4-9, we are shifting gears. Up until now, we have mastered the art of taking derivatives (finding the slope of a tangent line). Now, we are going to learn how to do the reverse: finding the original function given its derivative. This process is called finding the Antiderivative or Integration.

1. The General Antiderivative and the Arbitrary Constant

If you have a function $f(x)$, the antiderivative is a function $F(x)$ such that $F'(x) = f(x)$. However, there is a catch. Because the derivative of any constant number is zero, we must include an arbitrary constant, denoted as $C$, in our answer.

For example, if we ask "what did we differentiate to get $2x$?", the answer could be $x^2$, but it could also be $x^2 + 3$ or $x^2 - 100$. Therefore, the most general antiderivative is written as:

$$F(x) + C$$

Visually, as shown in the class notes, this represents a whole family of curves that are vertical shifts of one another.

2. Essential Integration Rules

To become proficient at integration, you need to recognize the reverse of your differentiation rules. Here are the heavy hitters from this lecture:

  • The Power Rule for Integration: For any $n \neq -1$: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ Tip: Add one to the exponent, then divide by the new exponent.
  • The Logarithm Rule: The special case where $n = -1$: $$\int \frac{1}{x} \, dx = \ln|x| + C$$
  • Trigonometry:
    • $\int \cos(x) \, dx = \sin(x) + C$
    • $\int \sin(x) \, dx = -\cos(x) + C$ (Watch the negative sign!)
    • $\int \sec(x)\tan(x) \, dx = \sec(x) + C$
  • Inverse Trigonometry: A key form to recognize from the notes is: $$\int \frac{1}{1+x^2} \, dx = \tan^{-1}(x) + C$$

3. Algebraic Strategy: Simplify First

Unlike differentiation, we don't have a simple "Quotient Rule" for integration yet. If you are faced with a fraction like the one in Example 1 ($g'(x) = \frac{2x^5 - \sqrt{x}}{x}$), you must use algebra to simplify before integrating.

Strategy: Split the fraction and convert roots to fractional exponents.

$$ \frac{2x^5}{x} - \frac{x^{1/2}}{x} = 2x^4 - x^{-1/2} $$

Once simplified, apply the Power Rule to each term individually.

4. Initial Value Problems (Finding C)

Sometimes, we are given specific information that allows us to find the exact value of $C$. This is called an Initial Value Problem.

  1. Find the general antiderivative (include $+C$).
  2. Plug in the given point (e.g., $f(0) = -2$).
  3. Solve the equation for $C$.
  4. Write the final specific function.

In the notes, we solved a second-derivative problem where we had to integrate twice, moving from $f''(x) \to f'(x) \to f(x)$. This required finding two constants, $C$ and $D$, using two different boundary conditions.

5. Application: Particle Motion

Integration allows us to move up the ladder of motion. If we know the acceleration of a particle, we can find its velocity and position:

  • Acceleration $a(t)$: Integrate to get Velocity.
  • Velocity $v(t)$: Integrate to get Position.
  • Position $s(t)$: The final function.

Remember: You will need initial conditions, like initial velocity $v(0)$ and initial displacement $s(0)$, to solve for the constants at each step.

Keep practicing your integration tables! Recognizing these patterns quickly is the key to success in Chapter 5. Happy studying!