Integral Calculator

The integral calculator computes definite and indefinite integrals of a function symbolically, applying the power rule, linear substitution, and standard antiderivative tables across polynomial, trigonometric, exponential, and logarithmic functions. An indefinite integral yields the antiderivative as a function; a definite integral evaluates that antiderivative between a lower bound and an upper bound to produce a single numeric value.

• Function notation follows standard convention: x^2 for powers, sin(x) for trigonometric functions, e^x or exp(x) for exponentials, ln(x) or log(x) for natural logarithms, sqrt(x) for square roots, and * for multiplication.
• Integration can be carried out with respect to any single variable, not only x, such as t, u, or theta.
• Bounds for a definite integral accept decimals, fractions (such as 3/4), mixed numbers (such as 1 1/2), and scientific notation (such as 1.5e3).
• Decimal precision is adjustable from 2 to 10 places, and a step-by-step solution can be shown to display each stage of the integration.
• Supported function forms include polynomial terms (ax^n), trigonometric functions (sin, cos, tan, sec, csc, cot), inverse trigonometric functions, exponential functions (e^x, a^x), logarithmic functions (ln(x)), square roots (sqrt(x)), and constant multipliers.
• When a function has no closed-form antiderivative, the calculator falls back to adaptive Simpson quadrature, a numerical method that approximates the definite integral to the selected precision.

Integrating f(x) = x^2 by the power rule gives the antiderivative x^3/3. As a definite integral from 0 to 2, that is (2^3)/3 - (0^3)/3 = 8/3 - 0 = 8/3, or approximately 2.6667 at four decimal places.

Integration is fundamental across mathematics, science, engineering, and economics, and this calculator covers a broad spectrum of integration problems from academic study to professional practice.

• Calculus Students: Verify homework answers and exam preparation by checking indefinite integrals of polynomial, trigonometric, and exponential functions. Entering an expression like x^3 - 2*x + 5 gives an antiderivative to compare against manual work.
• Physics and Engineering: Compute definite integrals for problems involving displacement from velocity functions, work done by variable forces, or electric flux calculations. Evaluating an integral such as sin(x) from 0 to pi confirms analytical results.
• Statistics and Probability: Calculate areas under probability density functions and cumulative distribution values by evaluating definite integrals over specified intervals.
• Economics and Business: Determine consumer surplus, producer surplus, or total revenue by integrating demand and supply functions over relevant price or quantity ranges.
• Research and Publication: Verify integral computations that appear in research papers or technical reports, using the step-by-step output to confirm that intermediate results match expected derivations.
• Teaching and Tutoring: Generate worked examples for classroom demonstrations or tutoring sessions. The step-by-step feature lays out clear explanations that help students understand integration techniques and rules.
• Numerical Analysis: Evaluate definite integrals of complex functions with no closed-form antiderivative using the built-in adaptive numerical integration method, accurate to the selected precision level.