The idea

Integration by reversing the chain rule

Spotting integrals of the form ∫f'(x)[f(x)]ⁿ, ∫f'(x)/f(x) → ln|f(x)| and ∫f'(x)e^{f(x)} — "integration by recognition", where the derivative of the inside appears as a factor.

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Pure · Integration

Integration by reversing the chain rule

Spotting integrals of the form ∫f'(x)[f(x)]ⁿ, ∫f'(x)/f(x) → ln|f(x)| and ∫f'(x)e^{f(x)} — "integration by recognition", where the derivative of the inside appears as a factor.

Why it works

The chain rule, backwards

Differentiation is mechanical; integration is recognition — and the chain rule leaves a fingerprint to recognise. It says

\dfrac{d}{dx}\,g(f(x)) = g'(f(x))\,f'(x)

— the derivative of the outer function times the derivative of the inside. Run that backwards and a powerful pattern appears: if the derivative of the inside function is sitting in the integrand as a factor, you can integrate by recognition.

Three patterns to know on sight

A power of a function: $\displaystyle\int f'(x)\,[f(x)]^n\,dx = \frac{[f(x)]^{n+1}}{n+1} + c$ $(n \ne -1)$ .
The $n = -1$ case — a log: $\displaystyle\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c$ . When the top is the derivative of the bottom, the integral is the log of the bottom. For example $\int \dfrac{2x}{x^2 + 1}\,dx = \ln(x^2 + 1) + c$ .
An exponential of a function: $\displaystyle\int f'(x)\,e^{f(x)}\,dx = e^{f(x)} + c$ . For example $\int \cos x\,e^{\sin x}\,dx = e^{\sin x} + c$ .

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