Sum and power rules
Integrate each term with the power rule and keep the constants outside. Each exponent goes up by one, and you divide by the new exponent.
Integral Calculator
Find indefinite and definite integrals with a step-by-step solution. Type a function, check how it is read, and see the answer, the rules used, and a graph.
Result
Checked: differentiating this answer gives back your function.
Step-by-step solution
Power rule
∫ uⁿ du = uⁿ⁺¹ / (n + 1), with n = .
Graph
The graph appears here once there is a valid function.
What is an integral?
Integration is the reverse of differentiation. If the derivative of a function F(x) is f(x), then F(x) is an antiderivative of f(x), and we write ∫ f(x) dx = F(x) + C. Integrals also measure accumulation: the definite integral of f between two limits is the signed area between the graph of f and the x-axis over that interval. This free integral calculator finds both kinds, shows the steps it used, checks its own answer, and graphs the function.
How to use the integral calculator
- Choose the type. Pick Indefinite for an antiderivative, or Definite to evaluate between two limits.
- Type your function. Use x as the variable, for example
x^2 + 3x. The keys under the box insert common symbols and functions. - Check the preview. The line "We read this as" shows exactly how your entry was understood, so you can spot a missing bracket early.
- Add limits for a definite integral. Numbers and expressions such as
pi/2orsqrt(2)both work. - Read the result. You get the answer, a step-by-step solution, and a graph. For a definite integral the area is shaded.
Indefinite and definite integrals
| Indefinite integral | Definite integral | |
|---|---|---|
| Notation | ∫ f(x) dx | ∫ₐᵇ f(x) dx |
| Result | A function, F(x) + C | A single number |
| Meaning | All antiderivatives of f | Signed area between a and b |
| Constant C | Included | Cancels out |
Basic integration rules
- Constant multiple
- ∫ k·f(x) dx = k·∫ f(x) dx
- Sum and difference
- ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
- Power rule
- ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, for n ≠ −1
- Reciprocal
- ∫ 1/x dx = ln|x| + C
- Exponential
- ∫ eˣ dx = eˣ + C, and ∫ aˣ dx = aˣ / ln(a) + C
- Substitution
- ∫ f(g(x))·g′(x) dx = ∫ f(u) du, with u = g(x)
- Integration by parts
- ∫ u dv = u·v − ∫ v du
- Fundamental theorem
- ∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f
Common integrals
These results come straight from the calculator, so you can compare them with your own work.
| Integral | Result |
|---|---|
Integration techniques with examples
Substitution
Let u = x² + 1, so du = 2x dx. The integral becomes ∫ 1/u du, which is ln|u|. Then replace u with x² + 1.
Integration by parts
Choose u = x and dv = eˣ dx. Then ∫ u dv = u·v − ∫ v du gives x·eˣ − ∫ eˣ dx.
Trigonometric powers
Let u = cos(x), so du = −sin(x) dx. The integral becomes −∫ u³ du, which is −u⁴/4.
Fractions with a quadratic denominator
Complete the square in the denominator and use the arctangent formula.
Example of a definite integral
To find the area under one arch of the sine curve, integrate sin(x) from 0 to π. The antiderivative is −cos(x), so the value is −cos(π) − (−cos(0)) = 1 + 1 = 2.
What this calculator can and cannot do
The calculator works with one variable, x, and handles these kinds of functions:
- Polynomials, powers, roots, and reciprocals, such as x³, √x, and 1/x².
- Exponential and logarithmic functions, such as eˣ, 2ˣ, and ln(x).
- Trigonometric, inverse trigonometric, and hyperbolic functions, including many powers and products.
- Rational functions with a linear or quadratic denominator, including polynomial long division.
- Integrals that need substitution or integration by parts, in the common cases.
Some integrals have no elementary antiderivative, such as e^(x²) or sin(x)/x, and some need methods that are not built in yet, such as partial fractions with higher-degree denominators, trigonometric substitution, or double integrals. In those cases the calculator says so and, for a definite integral, gives a numerical value. It does not guess.
Tips for working with integrals
- Add + C to every indefinite integral.
- Check an antiderivative by differentiating it. You should get the original function back.
- In this calculator,
log(x)means the natural logarithm, the same asln(x). - Use brackets for fractions:
1/(2x)is not the same as1/2x. - Look for breaks. A function such as 1/x cannot be integrated across x = 0.
Integral terms explained
- Integrand
- The function being integrated, the part between the integral sign and dx.
- Antiderivative
- A function whose derivative is the integrand.
- Constant of integration (C)
- The arbitrary constant added to an antiderivative, because constants vanish when you differentiate.
- Limits of integration
- The lower and upper values of x that bound a definite integral.
- Definite integral
- An integral with limits. Its value is a number.
- Improper integral
- An integral where the function is unbounded at a limit or the interval is infinite.
Frequently asked questions
What is an integral?
An integral is the reverse of a derivative. The indefinite integral of a function f(x) is a function F(x) whose derivative is f(x). The definite integral of f between two limits gives a number, the signed area between the graph of f and the x-axis over that interval.
What is the difference between a definite and an indefinite integral?
An indefinite integral has no limits and gives a family of functions, written F(x) + C. A definite integral has a lower and an upper limit and gives a single number. You find it by evaluating the antiderivative at the upper limit and subtracting its value at the lower limit.
Why do you add "+ C" to an indefinite integral?
The derivative of a constant is zero, so many functions that differ only by a constant have the same derivative. For example, x² + 1 and x² − 7 both have the derivative 2x. The constant of integration C stands for all of these possibilities. It is not needed for definite integrals, because the constant cancels when you subtract.
How do I type a function into the calculator?
Use x as the variable and type the function the way you would on a calculator: x^2 for x squared, 2x or 2*x for 2 times x, sqrt(x) for a square root, e^x for e to the power x, and ln(x) for the natural logarithm. Constants include pi and e, and the usual functions work, such as sin(x), cos(x), tan(x), and atan(x). Use brackets to make the order clear, for example 1/(2x). The preview under the input shows exactly how your entry is read.
Does the calculator show the steps?
Yes. It lists each rule it applies, such as the power rule, substitution, or integration by parts, with the integral before and after each step. It also checks every answer by differentiating it numerically and comparing the result with your function.
Why did the calculator say it could not find an antiderivative?
Some functions have no antiderivative that can be written with elementary functions. Examples are e^(x²) and sin(x)/x. Other integrals need techniques this calculator does not cover yet. For a definite integral, the calculator still gives a numerical value in these cases, as long as the integral converges.
Can a definite integral be negative?
Yes. A definite integral measures signed area. Area below the x-axis counts as negative, and swapping the limits changes the sign. If you want the total area between a curve and the axis, split the interval where the function changes sign and add the absolute values.
What happens when the function blows up between the limits?
If the function is undefined or unbounded inside the interval, such as 1/x from −1 to 1, the integral diverges or is undefined, and the calculator says so instead of giving a misleading number. If the function is unbounded only at an endpoint but the area is finite, such as 1/√x from 0 to 1, the calculator reports the value as an improper integral.