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器 · Integrals
Scroll of the Vessel
An integral fills a vessel: it measures the area under a curve. It is also the derivative run backwards.
Reverse the Edge
Find the function whose derivative is the one you have. Never forget the + C.
∫ cos(x) dx
- → Which function has derivative cos(x)?
- → sin(x)
Answer: sin(x) + C
Practice: Try ∫ 2x dx and ∫ 3x² dx.
Power Up
For xⁿ: raise the power by 1 and divide by the new power.
∫₁⁴ 1/x² dx
- → 1/x² = x⁻² → −x⁻¹ = −1/x
- → (−1/4) − (−1/1) = 3/4
Answer: 3/4
Practice: Try ∫₀² x dx (answer 2).
Mirror Cancel
For an odd function like x from −a to a, the area below the axis cancels the area above it.
∫₋₄⁴ x dx
- → Left half is negative, right half positive
- → They cancel
Answer: 0
Practice: Try ∫₋₂² x³ dx.
Into the dojo
Practise with the Vessel ∫ technique in Kung Fu Math from blue belt.
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