Mathematics · Calculus · Integral Calculus
Integral Calculator
Compute definite and indefinite integrals with complete step-by-step derivations. Symbolic engine handles linearity, power rule, linear substitution, u-substitution, integration by parts, and standard antiderivatives. Definite integrals fall back to adaptive Simpson quadrature when symbolic integration fails. Bounds may include π, e, and ±∞.
Formula
Integration is the reverse process of differentiation. The indefinite integral returns the family of antiderivatives F(x) + C; the definite integral computes the signed area under f from a to b via the Fundamental Theorem of Calculus: .
Integral Table
61 standard antiderivatives covering elementary functions.
Constant
f(x)
∫ f(x) dx
Power rule
f(x)
∫ f(x) dx
Reciprocal
f(x)
∫ f(x) dx
Square root
f(x)
∫ f(x) dx
Cube root
f(x)
∫ f(x) dx
Reciprocal squared
f(x)
∫ f(x) dx
Natural exp
f(x)
∫ f(x) dx
General exp
f(x)
∫ f(x) dx
exp(ax)
f(x)
∫ f(x) dx
Natural log
f(x)
∫ f(x) dx
x·ln(x)
f(x)
∫ f(x) dx
1/(ax+b)
f(x)
∫ f(x) dx
Sine
f(x)
∫ f(x) dx
Cosine
f(x)
∫ f(x) dx
Tangent
f(x)
∫ f(x) dx
Cotangent
f(x)
∫ f(x) dx
Secant
f(x)
∫ f(x) dx
Cosecant
f(x)
∫ f(x) dx
sec²
f(x)
∫ f(x) dx
csc²
f(x)
∫ f(x) dx
sec·tan
f(x)
∫ f(x) dx
csc·cot
f(x)
∫ f(x) dx
sin²
f(x)
∫ f(x) dx
cos²
f(x)
∫ f(x) dx
sin(ax)
f(x)
∫ f(x) dx
cos(ax)
f(x)
∫ f(x) dx
Arcsin
f(x)
∫ f(x) dx
Arccos
f(x)
∫ f(x) dx
Arctan
f(x)
∫ f(x) dx
Arctan (general)
f(x)
∫ f(x) dx
Arcsin (general)
f(x)
∫ f(x) dx
∫arcsin
f(x)
∫ f(x) dx
∫arccos
f(x)
∫ f(x) dx
∫arctan
f(x)
∫ f(x) dx
Hyp. sine
f(x)
∫ f(x) dx
Hyp. cosine
f(x)
∫ f(x) dx
Hyp. tangent
f(x)
∫ f(x) dx
Hyp. cotangent
f(x)
∫ f(x) dx
Hyp. secant
f(x)
∫ f(x) dx
sech²
f(x)
∫ f(x) dx
Inv. hyp. sine form
f(x)
∫ f(x) dx
Inv. hyp. cosine form
f(x)
∫ f(x) dx
x·eˣ
f(x)
∫ f(x) dx
x²·eˣ
f(x)
∫ f(x) dx
x·sin(x)
f(x)
∫ f(x) dx
x·cos(x)
f(x)
∫ f(x) dx
x²·sin(x)
f(x)
∫ f(x) dx
x²·cos(x)
f(x)
∫ f(x) dx
eˣ·sin(x)
f(x)
∫ f(x) dx
eˣ·cos(x)
f(x)
∫ f(x) dx
ln²(x)
f(x)
∫ f(x) dx
1/√(2ax−x²)
f(x)
∫ f(x) dx
Gaussian
f(x)
∫ f(x) dx
√(a²−x²)
f(x)
∫ f(x) dx
√(a²+x²)
f(x)
∫ f(x) dx
√(x²−a²)
f(x)
∫ f(x) dx
1/√(x²+a²)
f(x)
∫ f(x) dx
1/√(x²−a²)
f(x)
∫ f(x) dx
sec³
f(x)
∫ f(x) dx
ln³(x)
f(x)
∫ f(x) dx
x³·eˣ
f(x)
∫ f(x) dx
How it works
The engine performs recursive symbolic integration: it walks the expression tree and, at each node, attempts a sequence of strategies in priority order — constant rule, linearity, power rule, standard table lookup, linear substitution , heuristic u-substitution (find a sub-expression u(x) whose derivative appears as a factor), and integration by parts with iterative reduction for patterns.
For definite integrals, when the symbolic antiderivative is found the engine applies the Fundamental Theorem of Calculus: . Improper integrals (with bounds at ) are handled by progressive evaluation at increasingly large finite values until the result converges.
When symbolic integration fails to find a closed form, the engine falls back to adaptive Simpson quadrature on the definite interval. Numerical results are checked against common symbolic constants (π, e, simple fractions) and emitted symbolically when possible.
Worked example
Compute using integration by parts.
Let , . Then and:
The remaining integral is solved by parts again, yielding:
Type x^2*sin(x) in the calculator above to see the full derivation.
Frequently asked questions
What integration techniques does the calculator support?
Power rule, linearity, linear substitution f(ax+b), heuristic u-substitution, integration by parts (iterative for x^n·sin/cos/exp), reciprocal rule for 1/x and 1/(ax+b), arctan rule for 1/(1+x²), and standard antiderivatives for sin, cos, tan, sec, csc, exp, ln, sqrt, sinh, cosh and their inverses.
How are definite integrals computed?
When symbolic integration succeeds, the calculator applies the Fundamental Theorem of Calculus: F(b) − F(a). For improper integrals (bounds at ±∞), the antiderivative is evaluated at progressively larger finite values until convergence. If symbolic integration fails, adaptive Simpson quadrature provides a numerical answer.
Can I use π, e, and ∞ in the bounds?
Yes. Bounds accept any expression that evaluates to a number, including π (type "pi"), e, sqrt(2), pi/2, etc. For improper integrals, use "inf" or "-inf" for ±∞.
What is the constant of integration C?
Indefinite integration recovers an antiderivative only up to a constant: if F'(x) = f(x), then so is F(x) + C for any constant C. The calculator always appends '+ C' to the result of an indefinite integral. For definite integrals the constant cancels in the difference F(b) − F(a).
Why does the calculator return a numerical answer for some integrals?
Some integrals (like ∫e^(-x²)dx, the Gaussian) have no closed form in terms of elementary functions. For definite versions, adaptive Simpson quadrature provides high-precision numerical answers (typically 10⁻⁸ relative error). The result is checked against common constants like π and √π in case it matches a symbolic value.
Last updated: 2026-04-06 · Symbolic integration engine, written from scratch.