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Pure · Integration

Chapter 1 · 4

The idea

Indefinite integration (reversing differentiation)

What integration actually is — running differentiation backwards — and why the power rule for integration ("raise the power, divide by the new power") is the exact undo of the differentiation rule.

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Pure · Integration

Indefinite integration (reversing differentiation)

What integration actually is — running differentiation backwards — and why the power rule for integration ("raise the power, divide by the new power") is the exact undo of the differentiation rule.

Why it works

The reverse question

You know that y=x4+x2y = x^4 + x^2 has gradient 4x3+2x4x^3 + 2x. Now flip it: someone hands you the gradient 4x3+2x4x^3 + 2x and asks which function it came from. That is integration. Differentiation asks: "here's a function — what's its gradient?" Integration asks: "here's a gradient — what function has it?"

Undo both steps, in reverse order

Differentiating xnx^n does two things, in this order: multiply by the power, then knock the power down by one. To undo a two-step process you reverse both steps and their order — like taking off shoes and socks. So integrating does: raise the power by one first, then divide by the new power:

∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

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