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.

Pure · Integration

Indefinite integration (reversing differentiation)

Why it works

Integration answers the reverse question. Differentiation asks: "here's a function — what's its gradient?" Integration asks: "here's a gradient — what function has it?"

That's why the rule looks like differentiation run backwards. Differentiating

x^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:

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

Why divide by the new power? Try skipping it: differentiate

x^4

and you get

4x^3

— that factor of 4 appeared because the chain of differentiation multiplies by the power that was up there. If you wanted to end up at plain

x^3

, you must pre-shrink by that factor:

\frac{x^4}{4}

differentiates to exactly

x^3

. The division isn't a ritual — it cancels, in advance, the multiplication that differentiation is about to do.

This gives you a superpower no other A-level topic has so cleanly: integration is self-checking. Differentiate your answer. If you don't get back the thing you integrated, you're wrong — no answer book needed.

One practical consequence: the rule only speaks the language of powers. So

\sqrt{x}

\frac{1}{x^3}

and friends must be rewritten as powers (

x^{1/2}

x^{-3}

) before the rule can touch them. Most "integration errors" on these are really index-notation errors in disguise.