Integral Calculator
- Write the integral: ∫ x² + sin(x) dx.
- Methods needed: Sum rule, power rule, trigonometric rules.
- Find the antiderivative: F(x) = x³/3 − cos(x).
- Subtract: F(π) − F(0) = π³/3 + 2 ≈ 12.33542556.
- Sum rule: integrate each term separately and add the results.
- Power rule: the integral of xⁿ is xⁿ⁺¹/(n + 1) for n ≠ −1.
- Trigonometric rules: sin gives −cos, cos gives sin, and sec² gives tan.
- Adaptive numerical integration gives 12.33542556, which matches the exact value.
- Differentiating the antiderivative gives back f(x) at many sample points.
Integral Calculator: Definite and Indefinite Integrals with Steps
This free integral calculator finds the antiderivative of a function and, when you give it limits, the exact and decimal value of the definite integral. It handles polynomials, fractions, roots, trigonometric, exponential and logarithmic functions, and it shows the rules it used and the working. A graph shades the area under the curve so you can see what the number means.
Quick answer: the integral of x² + sin(x) from 0 to π is π³/3 + 2, which is about 12.3354. Its antiderivative is x³/3 − cos(x) + C.
What Is an Integral?
An integral adds up a quantity that keeps changing. If a function gives the speed of a car at each moment, its integral over an hour is the distance travelled. On a graph, the definite integral from a to b is the area between the curve and the x-axis. Integration is also the reverse of differentiation: the integral of a function is a new function whose derivative is the one you started with. The derivative calculator works in the opposite direction.
Definite and Indefinite Integrals
An indefinite integral has no limits and gives a family of functions that differ by a constant, which is why + C is added. A definite integral has a lower and an upper limit. It is found by taking the antiderivative, putting in the upper limit, and subtracting the value at the lower limit. The result is a number. Choose Definite or Indefinite at the top of the calculator, and the limit boxes appear only when they are needed.
Common Integration Rules
Most integrals are built from a short list of results. The power rule raises the exponent by one and divides by the new exponent. The sum rule lets you integrate a long expression one term at a time, and a number multiplying a function stays in front. When a function sits inside another one, substitution replaces the inner part with a single letter. When two different kinds of function are multiplied, integration by parts often works. Fractions made from polynomials are split into simpler pieces with partial fractions. Some examples are listed here.
| Function | Integral | Method |
|---|---|---|
| xⁿ | xⁿ⁺¹/(n + 1) + C | Power rule |
| 1/x | ln|x| + C | Reciprocal rule |
| e^(2x) | e^(2x)/2 + C | Exponential rule |
| sin(x) | −cos(x) + C | Trigonometric rule |
| 2x e^(x²) | e^(x²) + C | Substitution |
| x e^x | x e^x − e^x + C | Integration by parts |
| ln(x) | x ln(x) − x + C | Integration by parts |
| 1/(x² + 1) | atan(x) + C | Standard form |
How to Enter a Function and Limits
Type the function the way you would write it on paper. Use ^ for powers, so x squared is x^2, and write square roots as sqrt(x) and the exponential function as e^x. Functions such as sin, cos, tan and ln need brackets around their input. The shortcut buttons under the box insert common symbols for you. Limits can be plain numbers, or simple expressions like pi, pi/2 or sqrt(2). To integrate over an infinite range, type inf or -inf.
Area Under a Curve
A definite integral counts area above the x-axis as positive and area below it as negative, so the two can cancel. The calculator therefore also gives the Total area, which treats every part as positive, and the Average value of the function over the interval. These are useful in physics, engineering and economics, for example to find the total energy used, the work done by a changing force or the average cost over a period.
When There Is No Closed Form
Some functions, such as e^(−x²) or sin(x)/x, have no antiderivative that can be written with ordinary functions. For these, the indefinite integral cannot be shown, but the definite integral still has a value. Switch to Definite, enter the limits and the calculator finds the number with numerical integration. For related work, the polynomial calculator, the scientific calculator and the math solver can help with the steps around it.
How to Use This Integral Calculator
Choose the type
Pick Definite for a number with limits, or Indefinite for the general antiderivative.
Type your function
Enter it in the box, or use the shortcut buttons for powers, roots and functions.
Set the limits
For a definite integral, type or slide the lower and upper limits, such as 0 and pi.
Read the results
See the exact answer, the decimal value, the working and the shaded graph.