Integral Calculator

Compute symbolic integrals, explore areas under curves, and visualize antiderivatives. Powered by nerdamer CAS and high‑precision numeric methods.

Use ^ for exponents, * for multiplication, log() for natural log, exp() for exponential, sqrt(). Standard trig functions: sin, cos, tan, sec, csc, cot.
Quick examples:
x²
sin(x)
cos(x)
eˣ
ln(x)
1/x
x³ - 3x
x·sin(x)
Gaussian
Privacy-first computation: All symbolic integration and numeric evaluation happen locally in your browser using nerdamer CAS. No data is sent to any server.

What Is an Integral? From Accumulation to Antiderivatives

Integration is the fundamental tool for computing accumulation — areas under curves, total growth, probability, work, and more. The definite integral ∫ab f(x) dx represents the signed area between f(x) and the x‑axis from x = a to x = b. The indefinite integral (antiderivative) F(x) satisfies F'(x) = f(x) and introduces the constant of integration C.

Intuitive understanding: Integration is like "continuous summation." Imagine recording a car's velocity over time; the integral of the velocity function gives the total distance traveled. In economics, the integral of marginal cost equals total cost. Integration is the mathematical tool that accumulates infinitesimal changes into a total quantity.

\[ \int_{a}^{b} f(x)\,dx = F(b) - F(a) \quad \text{(Fundamental Theorem of Calculus)} \]

Our calculator automates both symbolic antidifferentiation and numeric quadrature.

Integration Strategy Guide

Different types of functions require different integration approaches:

  • Polynomials: Use the power rule directly: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
  • Trigonometric functions: Memorize basic forms: ∫sin(x)dx = -cos(x) + C, ∫cos(x)dx = sin(x) + C
  • Exponential functions: ∫eˣdx = eˣ + C, ∫aˣdx = aˣ/ln(a) + C
  • Rational functions: May require polynomial long division or partial fraction decomposition
  • Non‑elementary functions: Functions like ∫e^(-x²)dx have no elementary antiderivative; use numeric integration

Step‑by‑Step: Exploring an Integral with the Tool

Let's integrate f(x) = 3x² from 0 to 2:

  1. Enter "3*x^2" in the function field.
  2. Select "Definite Integral", set lower bound = 0, upper bound = 2.
  3. Click "Compute Integral & Plot". The tool returns the exact value 8 (since x³ from 0 to 2 = 8). The graph shades the area under the curve.
  4. Switch to "Indefinite Integral" to get the antiderivative: x³ + C.
  5. Use the evaluation field to compute F(2) = 8, confirming the definite result.

This workflow bridges symbolic calculus with geometric intuition — essential for mastering integration techniques.

No data retention: Since all calculations are performed locally, your input functions, integration limits, and results never leave your browser. All temporary data is automatically cleared when you close the page, ensuring complete privacy.
Integration approach: The tool attempts symbolic integration first; if unsupported, it falls back to high‑accuracy adaptive Simpson quadrature for definite integrals. For indefinite integrals, only symbolic results are displayed.
? Transparency note: For definite integrals, the calculator first computes the exact antiderivative symbolically (F(b)-F(a)). If symbolic integration fails or returns a non‑elementary form, it automatically switches to numeric integration (adaptive Simpson, tolerance 1e-8). The displayed result is accurate to at least 8 decimal places. You may see a decimal approximation even when an exact fraction exists – this ensures reliability for all integrands.

Calculation Methods and Verification

This calculator employs a tiered calculation approach with multiple verification layers:

  1. First tier: Symbolic integration - Uses nerdamer's symbolic integration engine, which implements algorithms like the Risch algorithm and heuristics to find exact antiderivatives.
  2. Second tier: Exact numerical evaluation - If a symbolic antiderivative is found, it computes F(b)-F(a) exactly, avoiding cumulative error.
  3. Third tier: Adaptive numerical integration - When symbolic integration fails, adaptive Simpson's rule is used, dynamically adjusting partition density to ensure convergence within the specified tolerance (typically 1e-8).
  4. Fourth tier: Result verification - All results undergo sanity checks, including parity verification, dimensional consistency, and cross-validation with alternative methods when possible.

Why Definite Integrals Matter in Real Life

Engineers use integrals to calculate center of mass, fluid flow, and total energy. In economics, the integral of marginal cost gives total cost. Probability distributions rely on integrals to compute cumulative probabilities. Our calculator handles both elementary functions and more complex integrands using robust fallback methods.

Real‑World Application: Total Distance & Accumulation

Suppose a particle's velocity is v(t) = 3t² − 12t + 9 (m/s). The net displacement from t=0 to t=4 is ∫₀⁴ v(t) dt. The calculator instantly returns 4 meters. The total distance traveled (integrating |v(t)|) requires splitting intervals, but our tool quickly verifies definite integrals for displacement — an essential check for physics problems.

Integration Rules & Techniques — Quick Reference

Rule Integral Result
Power Rule ∫ xⁿ dx xⁿ⁺¹/(n+1) + C (n ≠ -1)
Natural Log ∫ 1/x dx ln|x| + C
Exponential ∫ eˣ dx eˣ + C
Trig (sine) ∫ sin(x) dx -cos(x) + C
Trig (cosine) ∫ cos(x) dx sin(x) + C
Integration by Parts ∫ u dv uv − ∫ v du
Substitution ∫ f(g(x))g'(x) dx F(g(x)) + C

Frequently Asked Questions

Our calculator supports polynomials, trigonometric (sin, cos, tan, sec, csc, cot), exponentials, logarithms, and many compositions. If symbolic integration fails, definite integrals fall back to adaptive numeric integration (Simpson's rule) with high precision.

For indefinite integrals we display the antiderivative followed by "+ C". Graphically, we plot the antiderivative with C = 0, showing one representative curve.

This version focuses on continuous elementary functions. For piecewise or |x|, consider splitting the domain manually. The definite integral numeric fallback may still provide an approximation.

Improper integrals with infinite limits are not directly supported, but you can approximate by setting large finite bounds. For symbolic improper integrals, use specialized CAS tools.

Yes, we implement adaptive Simpson quadrature with a tolerance of 1e-8. For well-behaved functions, results are accurate to at least 6 decimal places. Our numeric engine is based on proven numerical analysis algorithms and includes automatic error estimation.

When the symbolic antiderivative exists but evaluating F(b)-F(a) yields an irrational number (e.g., π, e, sqrt(2)), or when the integrand is non‑elementary, the calculator automatically uses numeric integration. The result is still highly accurate (up to 8 decimal places). For exact rational results, try functions like polynomials or simple trig where the antiderivative is elementary and evaluable symbolically.

Recommended verification steps:

  1. Differentiate the antiderivative: The derivative of the antiderivative F(x) should equal the original function f(x).
  2. Check boundary cases: For odd functions over symmetric intervals, the integral should be zero; for constant functions, the integral equals constant × interval length.
  3. Numerical cross‑check: Subdivide the interval and use simple methods (rectangular, trapezoidal) to approximate the integral, comparing with the calculator's result.
  4. Dimensional analysis: In physical applications, verify that the integral's dimensions are consistent with expected physical quantities.

The calculator internally implements multiple verification mechanisms, including symbolic‑numeric cross‑validation, to ensure result reliability.

Academic foundation: Built upon nerdamer's symbolic integration engine and verified against standard calculus textbooks (Stewart, Thomas). Our numeric integration engine implements adaptive Simpson's rule – a proven high‑accuracy numerical integration technique that automatically adjusts partition density based on function variation. This algorithm, derived from classical numerical analysis literature (e.g., Burden and Faires' Numerical Analysis), has been optimized for the web environment. All results undergo cross‑validation: for test functions with known analytic solutions, both symbolic and numeric integration are run simultaneously to ensure numeric results match analytic solutions within tolerance. Reviewed by mathematics educators to ensure reliability and pedagogical value. Updated April 2026.

References: Wolfram MathWorld – Integral; nerdamer CAS documentation; OpenStax Calculus Volume 2; Burden, R.L. and Faires, J.D., "Numerical Analysis".