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Pure · Integration

Chapter 1 · 3

The idea

Integrating standard functions

The integrals that are just the derivative rules in reverse — e^x, 1/x → ln|x|, sin, cos and sec² — together with the f(ax+b) rule for a linear inside function (integrate as normal, then divide by a).

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Pure · Integration

Integrating standard functions

The integrals that are just the derivative rules in reverse — e^x, 1/x → ln|x|, sin, cos and sec² — together with the f(ax+b) rule for a linear inside function (integrate as normal, then divide by a).

Why it works

Derivatives, read backwards

Integration is differentiation run backwards, so every derivative you know gives you an integral for free — just read the rule the other way. Once you have the Year 2 derivatives of exe^x, ln⁡x\ln x and the trig functions, a whole table of standard integrals falls out: ∫ex dx=ex+c,∫cos⁡x dx=sin⁡x+c,∫sin⁡x dx=−cos⁡x+c,\int e^x\,dx = e^x + c, \qquad \int \cos x\,dx = \sin x + c, \qquad \int \sin x\,dx = -\cos x + c, ∫sec⁡2x dx=tan⁡x+c,∫sec⁡xtan⁡x dx=sec⁡x+c,∫cosec⁡2x dx=−cot⁡x+c.\int \sec^2 x\,dx = \tan x + c, \qquad \int \sec x\tan x\,dx = \sec x + c, \qquad \int \operatorname{cosec}^2 x\,dx = -\cot x + c.

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