Integrate

The symbolic integral of an expression, with optional bounds.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaIntegral↗MathWorldIntegral↗WikidataQ80091↗Britannicatopic/integral-mathematics↗Fungrim entrySageMathintegrate($*,)SymPyintegrate($*,)

Integrate(f, limits+)Adds a closed form for $\int \sin^m(u)\cos^n(u)\,dx$ with a linear argument $u = ax+b$ and $m$ or $n$ odd, expanding the odd power through $\sin^2+\cos^2=1$ into a polynomial antiderivative. Under N, over a semi-infinite interval, an oscillatory integrand that isn't finite at the finite end has its first lobe integrated without evaluating it there, where compute-engine drops the sliver next to the end; lobes that beat leave the integral unevaluated. Falls back to compute-engine's native integrator otherwise.

Typed signatures

Type	Package	Applies
(function, limits+) -> list<tuple> | number | tensor | tuple	analyticreplaces evaluation	always
(function, limits+) -> list<tuple> | number | tensor | tuple	compute-engine	always
(function, limits+) -> list<tuple> | number | tensor | tuple	evaluationreplaces compute-engine	always

Domain: Compute engine

Examples

Basic #

$\int \sin^3 x\,dx$

live ↗

In$\int\!\sin(x)^3\, \mathrm{d}x$Out$\frac{\cos(x)^3}{3}-\cos(x)$

$\int \sin^3 x \cos x\,dx$

live ↗

In$\int\!\sin(x)^3\cos(x)\, \mathrm{d}x$Out$\frac{\cos(x)^4}{4}-\frac{\cos(x)^2}{2}$

Scope #

$\int_0^\pi \sin x\,dx = 2$

live ↗

In$\int_{0}^{π}\!\sin(x)\, \mathrm{d}x$Out$2$

About

Examples	3
Checked against	Wolfram Language 3 agreeSageMath 3 agreeSymPy 3 agree
Crosswalk	WikipediaMathWorldWikidataBritannicaFungrim entry ×15SageMathSymPy

Implementation

mappedsympypackages/oracle/src/mappings.ts

Any number of iterators. A bare variable, or Limits(x, Nothing, Nothing), is an indefinite integral, which the scan compares up to a constant. An iterator is SymPy's (x, a, b). The first limit is the outermost, as Wolfram's, so the iterators are emitted in reverse; SymPy integrates its first iterator innermost.

mappedsagepackages/oracle/src/mappings.ts