β« f(x) dx β step by step
Type a function β the formula renders live. Press "=" and the calculator builds the antiderivative along with a full chain of reasoning: which rule applies at each step and why.
Solved it yourself? Enter your answer (for an indefinite integral you can include + C) β the calculator compares it against the original function, not a fixed answer, so any correct form is accepted.
No responsibility is taken for the correctness of results. Software errors can produce incorrect results β always double-check anything important.
The online integral calculator evaluates both indefinite and definite integrals and provides step-by-step solutions with the rules applied at each step. For definite integrals, it offers numerical evaluation, convergence checks, and an interactive graph.
The service finds antiderivatives symbolically, providing exact analytical formulas rather than approximate numerical results. It supports the main integration methods, including substitution, integration by parts, standard integrals of polynomial, trigonometric, hyperbolic, and exponential functions, as well as more advanced transformations.
If no elementary antiderivative can be found, the calculator does not present an approximate result as an exact solution. Instead, it switches to numerical integration when possible.
For definite integrals, the service checks the interval of integration for singularities. If the integral diverges, the calculator reports the divergence instead of returning a plausible but incorrect number.
Definite integrals also include an interactive graph with signed area shading. Positive and negative regions are displayed separately, making it easy to see how they cancel each other out.
The calculator also includes an answer-checking feature. It compares the submitted answer with the original integrand and accepts any mathematically equivalent form rather than requiring one specific representation.
The mathematical expression is first parsed syntactically. The calculator recognizes implicit multiplication (for example, 2sin(x) is interpreted as 2*sin(x)) and function aliases (tg, ctg, sh, and ch are recognized alongside tan, cot, sinh, and cosh), then converts the expression into a tree suitable for further symbolic transformations.
The computer algebra system then attempts to find an analytical antiderivative. It uses algorithms similar in principle to the Risch algorithm, applying and combining integration rules, substitutions, and transformations until a closed form is found or it becomes clear that the antiderivative cannot be expressed in elementary functions.
To provide more than just the final result, the service builds a tree of the rules applied β such as integration by parts, substitution, and factoring out constants β and converts it into a sequence of readable steps with mathematical formulas.
For definite integrals where symbolic computation does not produce a closed form, numerical integration using quadrature methods is applied. Before the calculation, the integration interval is checked for singularities: if the integrand tends to infinity within the interval, the service detects the case and reports that the integral diverges.
The graph of the function and its antiderivative is generated separately from the main calculation. For a definite integral, the region between the graph and the x-axis is shaded according to the sign of the function, illustrating the geometric meaning of the integral as signed area.
Enter the function you want to integrate without f(x) = and without the differential dx; neither needs to be included. The formula is rendered graphically as you type, so you can immediately check that the calculator has interpreted the expression as intended.
Use parentheses wherever they are needed to specify the order of operations, for example a/(b+c). Use a period rather than a comma for decimal numbers: 3.141, not 3,141. The examples section includes the most commonly entered functions and can be used as a reference for the syntax.
The integration variable and integration limits can be specified separately. If no limits are provided, the calculator finds only the antiderivative (an indefinite integral). If limits are provided, the service additionally calculates the numerical value of the definite integral and displays an interactive graph with the region between the function and the x-axis.
Once the expression has been entered, click β=β and the calculator will display the antiderivative, the step-by-step solution, and, if limits are specified, the value of the definite integral together with the graph.