1. Mirror the Trig: $\int \cos x = \sin x$, but $\int \sin x = -\cos x$.
2. Eternal e: $\int e^x dx = e^x + C$. (It never changes!).
3. The Log-Anchor: $\int \frac{1}{x} dx = \ln|x| + C$.
4. The Derivative Check: Always take the derivative of your answer to see if you get the original back.
Find the indefinite integral for each function.
$\int 5 \cos x dx$
$\int (e^x + \sin x) dx$
$\int \frac{4}{x} dx$
Use the Law of Linearity to integrate the complex expressions.
$\int (3x^2 + 2e^x - \cos x) dx$
$\int (10 - \sin x + \frac{1}{x}) dx$
Take the function $f(x) = e^x + 5$.
1. Differentiate it ($f'$).
2. Integrate the result ($\int f'$).
Did you get the 5 back, or did it become a $+C$? Why is the "5" hidden from the derivative?
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Find $\int (-2\sin x + \sec^2 x) dx$.
A sound wave has a rate of change $v(t) = \sin(t) - \cos(t)$.
Find the total displacement function $s(t) = \int v(t) dt$.
Objective: Explain the Unchanging $e$ to a younger sibling.
The Activity:
1. Stand in front of a mirror.
2. Wave. Your reflection waves back. (Inverse).
3. Smile. Your reflection smiles back.
4. "Some things in math are like this mirror. No matter what we do to them, they stay exactly the same. That's the number $e$."
The Lesson: "God's love is like the $e$. It's the same yesterday, today, and forever."
Response: ___________________________________________________________