Integrals
Antiderivatives with the power rule backwards, and definite integrals of polynomials.
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About Integrals
Integration undoes differentiation. The power rule runs backwards: add 1 to the exponent and divide by the new exponent. An antiderivative always comes with + C, because any constant disappears when you differentiate.
A definite integral has limits on it and gives a number, the signed area between the curve and the x-axis. Find an antiderivative, put in the top limit, put in the bottom limit, and subtract.
Keep in mind
- ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1.
- Integrate term by term; a constant k becomes kx.
- Definite integral: F(top) − F(bottom). No C needed, it cancels.
- Area below the x-axis counts as negative.
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