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 ^ for powers, * or juxtaposition for multiplication (2x means 2·x), and function names like sin, cos, ln, exp, and sqrt — for example x^2, 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 x^2, 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 ^ for powers, * or juxtaposition for multiplication, and function names like sin, cos, ln, exp, and sqrt.
Which integration rules does it use?
For indefinite integrals it applies linearity (integrate term by term), the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1), the reciprocal rule ∫1/x dx = ln|x|, exponentials ∫e^(ax+b) dx, and standard functions with a linear inner argument via u-substitution, such as ∫sin(ax+b) dx. It appends + C to indefinite results.
Does it do definite integrals and area under the curve?
Yes. Enter the limits a and b and the calculator evaluates the definite integral. When an elementary antiderivative F exists it uses the fundamental theorem, F(b) − F(a); otherwise it computes the value numerically with adaptive Simpson quadrature and labels it numerical. The graph shades the signed area between a and b.
What if there is no elementary antiderivative?
Some functions, like e^(−x²) or sin(x²), have no antiderivative in elementary terms. The calculator says so for the indefinite integral, but still returns an accurate definite value between your limits using numerical integration.