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Antiderivatives and indefinite integrals

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

What is it?

$F$ is an antiderivative of $f$ if $F' = f$ ; all of them differ by a constant, written $\int f(x)\,\dd x = F(x) + C$ . Many elementary functions, such as $e^{-x^2}$ , have no elementary antiderivative.

Formulas

\int x^n\,\dd x = \frac{x^{n+1}}{n+1} + C\ (n \ne -1), \quad \int\frac{\dd x}{x} = \ln|x| + C, \quad \int e^x\,\dd x = e^x + C

Where it shows up in computing

Symbolic computation (CAS)★★★★★fundamentalScientific computing and algorithms
The Risch algorithm decides whether an elementary antiderivative exists and finds it — a landmark of computer algebra.

Where is it used?

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

λ Scientific computing and algorithms

Symbolic computation (CAS)★★★★★
Fundamental theorem of calculus→Cumulative distribution function→Monte Carlo methods★★★★★
Ordinary differential equations→Euler's method→Runge–Kutta methods→Scientific computing★★★★★
Fundamental theorem of calculus→Improper integrals→Integral test→Algorithm analysis and complexity★★★★★

ℒ AI and machine learning

Ordinary differential equations→Neural ODEs★★★★★
Fundamental theorem of calculus→Improper integrals→Probability density function→Bayesian inference★★★★★
Fundamental theorem of calculus→Improper integrals→Probability density function→Generative models★★★★★
Fundamental theorem of calculus→Improper integrals→Probability density function→Expectation→Loss function★★★★★
Fundamental theorem of calculus→Improper integrals→Probability density function→Maximum likelihood estimation→Logistic regression★★★★★
Fundamental theorem of calculus→Improper integrals→Probability density function→Expectation→Stochastic gradient descent (SGD)★★★★★
+10

3D Computer graphics

Fundamental theorem of calculus→Cumulative distribution function→Monte Carlo methods→The rendering equation★★★★★

⚙ Robotics and control

Ordinary differential equations→Control theory★★★★★
Ordinary differential equations→Control theory→Trajectory optimization and MPC★★★★★
Fundamental theorem of calculus→Improper integrals→Probability density function→Continuous distributions→Kalman filter★★★★★
Ordinary differential equations→Laplace transform→PID control★★★★★
Ordinary differential equations→First-order linear equations→Systems of ODEs→Robot dynamics★★★★★

⚛ Physics and simulation

Ordinary differential equations→Classical mechanics★★★★★
Ordinary differential equations→Physics engines★★★★★
Ordinary differential equations→Population and epidemic models★★★★★
Integration by parts→Finite element method★★★★★
Ordinary differential equations→Physics engines→N-body gravitational simulation★★★★★
Ordinary differential equations→Partial differential equations→Heat equation and diffusion★★★★★
+3

∿ Signals, media and vision

Integration by substitution→Trigonometric integrals→Fourier series→Signal processing★★★★★
Fundamental theorem of calculus→Improper integrals→Fourier transform→Sampling theorem (Nyquist–Shannon)★★★★★
Fundamental theorem of calculus→Improper integrals→Fourier transform→Fast Fourier transform (FFT)★★★★★
Ordinary differential equations→Laplace transform→Z-transform→Digital filters★★★★★
Fundamental theorem of calculus→Improper integrals→Fourier transform→Telecommunications (modulation, OFDM)★★★★★
Integration by substitution→Trigonometric integrals→Fourier series→Signal processing→Media compression (JPEG, MP3, video)★★★★★
+1

⇄ Optimization and systems

Fundamental theorem of calculus→Improper integrals→Probability density function→Continuous distributions→Queueing theory and performance★★★★★
Ordinary differential equations→Euler's method→Gradient descent→Recommender systems★★★★★

⊕ Cryptography and security

Integration by substitution→Trigonometric integrals→Fourier series→Signal processing→Side-channel analysis★★★★★

ψ Quantum computing and physics

Fundamental theorem of calculus→Improper integrals→Fourier transform→Uncertainty principle★★★★★

What depends on it

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