Section 5.1
Indefinite Integrals
We know how to find the slope given a curve. Now we reverse the process: Given the slope, can we find the curve?
1
The Power of "+C"
The Indefinite Integral asks for the function whose derivative is .
d/dxvs∫
Why "+C"?
Because differentiation destroys constants ( and ). When we integrate, we must acknowledge that "lost" constant.
2
Worked Example: Initial Value Problem
Finding a specific curve
Find given slope and point .
The Family of Curves
Derivative $f'(x) = 2x-1$ is the same for all.
Drag the slider to choose a specific member of the family (Solving the Initial Value Problem).
The slope is always the same, but only ONE curve hits the target point.
1. General Solution
2. Initial Condition
Plug in :
3. Specific Solution
3
Level Up Examples
Example A: Trig Functions
Evaluate .
1. Reverse Power Rule:
Recall .So .
2. Reverse Trig:
Recall .So .
3. Result:
.Example B: Recognizing Derivatives
Evaluate .
1. Reverse Exponential:
Recall .So .
2. Reverse Trig:
Recall .So .
3. Result:
.5
Practice Quiz
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