Integration by substitution

Level FundamentalDifficulty ★★★★★Method⌖ Open in the map

What is it?

The chain rule read backwards: $\int f(g(x))\,g'(x)\,\dd x = \int f(u)\,\dd u$ . In probability it is how densities transform when a random variable is transformed.

Formulas

$\int_a^b f\big(g(x)\big)\,g'(x)\,\dd x = \int_{g(a)}^{g(b)} f(u)\,\dd u$; $Y = g(X) \implies p_Y(y) = p_X\big(g^{-1}(y)\big)\left|\frac{\dd g^{-1}}{\dd y}\right|$
change of variables for densities

Where it shows up in computing

Monte Carlo methods★★★★★frequentScientific computing and algorithms
Importance sampling and inverse-transform sampling are changes of variable in the integral being estimated.

Where it shows up in AI

Generative models★★★★★advancedAI and machine learning
In 1D, a normalizing flow is exactly the density change-of-variables formula.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

What is it?

Formulas

Where it shows up in computing

Where it shows up in AI

Where is it used?

⚛ Physics and simulation

∿ Signals, media and vision

⊕ Cryptography and security

ψ Quantum computing and physics

ℒ AI and machine learning

λ Scientific computing and algorithms

3D Computer graphics

What depends on it