Understand the **Common Integrals** that reverse the basic differentiation rules. Focus on the trigonometric functions ($\sin, \cos, \sec^2$) and the exponential function ($e^x$). Reflect on the **Symmetry of Truth**. For every way that God allows us to "Go Out" into the derivative, He has provided a way to "Come Back" through the integral. Meditate on the path of repentance—it is the inverse of the path of transgression.
In Phase 1, we learned that some things change in very specific ways.
- Sine changes to Cosine.
- Cosine changes to Negative Sine.
- And $e^x$... stays exactly the same.
The number **$e$** is the signature of Unchanging Life. Jesus said, "I am the resurrection and the life" (John 11:25). No matter how many times you differentiate $e^x$, its speed is always equal to its position. And no matter how many times you integrate it, it returns to itself.
$\\int e^x dx = e^x + C$
Today, we learn to reverse the "Language of the Shift." we will see that the trigonometric functions are a **Cycle of Praise**—rising and falling in an eternal loop. we will learn that even when our path feels like a complex wave, there is a simple "Back-Way" to the heart of the Father.
The Rupture: The student integrates $\sin x$ and writes $\cos x + C$.
The Repair: "Watchman, you have forgotten the **Cost of the Turn**! If you differentiate $\cos x$, you get $-\sin x$. If you claim the integral of $\sin x$ is $\cos x$, your derivative check will fail ($-\sin eq extrm{sin}$). To get a positive result, you must start with a negative intent. $\\int extrm{sin } x = -\cos x$. Check your signs with the 'Derivative Test' every time, or your restoration will be upside down."
1. **Apply the Rule**: Use the table to find the anti-derivative.
2. **The Constant**: Add $+C$ immediately.
3. **The Proof**: Differentiate your answer. If you don't get the *exact* original function, you made a sign or power error.
The older student should use a tape recorder or a video player. "Look at this video of me jumping ($f$). If I play it in slow motion, you can see my speed ($f'$). If I play it **Backwards** (Rewind), you see me returning to the floor where I started ($f + C$)."
The older student must explain: "In my math, the Integral is the 'Rewind' button. It takes the speed of the jump and shows us where we were standing at the beginning."
A person's breath rate follows the curve $R'(t) = extrm{cos}(t) + e^t$.
Task: Find the general formula for the total amount of air inhaled ($R(t)$).
Theological Requirement: Notice how the breath is a combination of a **Cycle** ($ extrm{cos}$) and an **Exponent** ($e^t$). Reflect on how our spiritual life is both a cycle of seasons and a constant expansion of grace. Why does God want us to be able to "Sum up" both kinds of change?
Why is the integral of $1/x$ written as $\ln|x|$?
Because the Natural Logarithm can only handle positive numbers. But $1/x$ can be negative. By using the **Absolute Value**, we are ensuring that the restoration works for every $x$ except zero.
This teaches us the **Law of the Absolute**. God's restoration is not limited by our "Negative" states. He applies the absolute value of His grace to our negative history, allowing us to find the "Abundant Rhythm" ($\\ln$) no matter where we were on the number line.
The "Derivative Check" is the most powerful tool in Phase 2. Encourage the student to **Never** turn in an integration problem without checking the derivative.
This builds a sense of **Logical Self-Sufficiency**. They don't need a teacher to tell them if they are right; the math itself confirms the truth.