Integral Calculator

Integrate any function of x — get the indefinite integral (antiderivative) with steps, or add limits a and b for the definite integral and the shaded area under the curve.

Indefinite integral —

How to Integrate a Function

Enter a function of x to see its integral and the rules behind it.

1

Write your function of x

Type f(x) using * or juxtaposition for multiplication (2x means 2·x), and function names like sin, cos, ln, exp, and sqrt — for example x², 1/x, or sin(2x).

2

The calculator integrates term by term

It applies linearity, the power rule, the reciprocal rule (∫1/x = ln|x|), exponentials, and u-substitution for a linear inner argument — showing each rule. For definite integrals, add the limits a and b.

3

Read the antiderivative, area, and graph

The indefinite integral appears with + C; with limits, you get the definite value and the shaded area under the curve. Going the other way? The Derivative calculator differentiates, and the Limit calculator evaluates limits.

Frequently Asked Questions

How do you use this integral calculator?
Type your function of x into the f(x) box — for example x², sin(x), or 1/x — to get the indefinite integral (antiderivative) with steps. To compute a definite integral, also fill in the lower limit a and upper limit b; the calculator returns the value and shades the area under the curve between a and b. Use * or juxtaposition for multiplication, and function names like sin, cos, ln, exp, and sqrt.
Which integration rules does it use?
For indefinite integrals the calculator applies linearity (integrating term by term), the power rule for every exponent except −1, the reciprocal rule ∫1/x dx = ln|x|, exponentials, and standard functions with a linear inner argument via u-substitution such as ∫sin(ax+b) dx. So ∫x³ dx = x⁴/4 + C and ∫sin(2x) dx = −cos(2x)/2 + C. Every indefinite result carries the + C, because an antiderivative is only determined up to a constant. The exponent −1 is the exception that makes the reciprocal rule a separate case.
Does it do definite integrals and area under the curve?
Definite integrals and shaded areas are both supported. Enter the limits a and b and the calculator evaluates the integral: when an elementary antiderivative F exists it uses the fundamental theorem, F(b) − F(a), and otherwise it computes the value numerically with adaptive Simpson quadrature and labels it as numerical. The integral of x² from 0 to 1 is returned exactly as 1/3. The graph shades the signed area between a and b, so a region below the axis subtracts rather than adds.
What if there is no elementary antiderivative?
Some functions, such as e−x² or sin(x²), have no antiderivative that can be written with elementary functions at all, which is a proven fact rather than a limitation of the software. The calculator says so for the indefinite integral instead of inventing an answer. It still returns an accurate definite value between your limits using adaptive numerical integration. The integral of e−x² from 0 to 1 is about 0.746824, a number only a numerical method can produce.

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