Integration as the Reverse of Differentiation
Recover a function from its derivative by raising the power and dividing, and carry the constant of integration that differentiation destroyed. Use a known point to pin that constant down.
What a learner can do afterwards
- Integrate 6x² - 4x and check the answer by differentiating it
- Explain why every answer carries + c
- Find the particular function whose derivative is 2x and which passes through (1, 5)
1 · Read
Last stop you built a function from a story and differentiated it. Every stop since the shrinking chords has run the same way: hand over a curve, get back its gradient. Now run it backwards. The Ridgeway tank has a flow meter on the outlet, so the crew knows dV/dt at every moment, and what they actually want is V. That is the reverse question, and it is called integration. One thing makes it easier than it sounds: you can always check an answer by differentiating it. If it gives back what you were handed, it is right.
Differentiating multiplies by the power and drops it by one, so reversing means raising the power by one and dividing by the new power. Written down: the integral of xⁿ is xⁿ⁺¹/(n + 1), plus a constant. A coefficient is divided by the new power too, so 5x³ gives 5x⁴/4. A sum is done term by term, exactly as before. A plain number k is really kx⁰, so it integrates to kx. The one power the rule cannot handle is n = -1, because dividing by n + 1 would be dividing by zero.
Integrate 6x² - 4x. Take it a term at a time. For 6x², the power goes from 2 to 3 and you divide by 3, so 6 becomes 2 and the term is 2x³. For -4x, remember it is -4x¹, so the power goes to 2 and you divide by 2, giving -2x². Then add the constant. The answer is 2x³ - 2x² + c. Now check it by differentiating: 2x³ gives 6x², and -2x² gives -4x, so it does give back what you started with.
That constant is not a decoration. Differentiating kills any constant, so x² + 1, x² + 7 and x² - 30 all have derivative 2x, and there is no way of telling from 2x alone which one you began with. Running the process backwards therefore recovers a whole family of curves, identical in shape and stacked at every possible height. Writing + c is how you say that honestly. The tank meter has the same problem: it saw the flow, but it never saw how full the tank started.
One extra fact picks one curve out of the family. Suppose dy/dx = 2x and the curve passes through (3, 11). Integrating gives y = x² + c, which is still the whole family. Now use the point: substituting x = 3 and y = 11 gives 11 = 9 + c, so c = 2 and the curve is y = x² + 2. Check it: at x = 3 that gives 9 + 2 = 11. For the tank, the point is the starting volume, and without it the meter can never say how much is left.
Integration undoes differentiation: raise the power by one and divide by the new power, term by term, dividing the coefficient too. Every answer carries + c, because differentiating destroys a constant and no derivative can say which one was there. That + c stands for a whole family of curves stacked at every height. One known point pins it down: substitute the coordinates, solve for c, and one curve is left. Check any answer by differentiating it back.
2 · Watch
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24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.