Integral calculator, with the method named.

An integral calculator finds the antiderivative of a function, which is a function whose derivative is the one you typed. Add two limits and it gives the area under the curve between them. This one works out the steps and names each method, so you can follow it. It covers polynomials, powers and roots, exponentials, logs, trig and inverse trig functions, products that need integration by parts, fractions that need partial fractions, and many substitutions.

How to use the integral calculator

1. Type the function in Function, for example x e^x. Leave out the dx. Multiplication can be left out.
2. Read the line that starts "Read as" to make sure it understood your brackets.
3. For an indefinite integral, leave Lower limit and Upper limit empty. For a definite integral, fill in both. Use numbers, pi, e, or inf and -inf.
4. Read the answer. An indefinite integral ends in + C. A definite integral shows an exact value when one exists, with a decimal beside it.
5. Open the steps under the answer. Use Copy answer or Copy link to keep it.

The methods, with one example each

• Power rule. The integral of x^n is x^(n+1)/(n+1) + C for any n except −1. So the integral of x^2 is x^3/3, and the integral of 1/x is ln|x|.
• Substitution. For x e^(x^2), let w = x^2. Then dw = 2x dx, so the integral becomes half the integral of e^w, which is e^w/2. Put x^2 back and the answer is e^(x^2)/2 + C.
• Integration by parts. For x e^x, take u = x and dv = e^x dx. Then v = e^x and the integral is x e^x − the integral of e^x, which is x e^x − e^x + C. The calculator repeats the step when the power of x is higher, so x^2 e^x takes two rounds.
• Partial fractions. For 1/(x^2 − 1), split it as 1/(2(x − 1)) − 1/(2(x + 1)). Each piece is a log, so the answer is (1/2) ln|x − 1| − (1/2) ln|x + 1| + C.
• Trig identities. For sin^2(x), use sin^2 = (1 − cos(2x))/2. The integral is x/2 − sin(2x)/4 + C.

Definite integrals, worked

A definite integral is F(upper) − F(lower), where F is the antiderivative. Here are four you can type in to check.

The last row is an improper integral, because the upper limit is infinity. The calculator pushes the limit outwards and checks that the answer settles on a number. If it does not settle, it says the integral diverges. It also looks for places between the limits where the function blows up, such as 1/x from −1 to 1, and tells you the integral has no finite value there instead of giving a wrong number.

When there is no answer

Some functions have no antiderivative that can be written with the usual functions. Two famous ones are e^(x^2) and sin(x)/x. When the calculator cannot find one, it says so. If you gave limits, it still gives a numeric estimate and labels it as an estimate, not an exact answer. A few functions do have an antiderivative that needs a method this calculator does not have yet, such as some products of three trig functions or square roots of general quadratics in a product.

Why there is a + C

Differentiating removes any constant, so many functions share the same derivative. x^2 + 1 and x^2 + 7 both give 2x. The + C stands for that unknown constant. A definite integral has no C, because the constant cancels when you subtract.

Common problems

• The answer can look different from a textbook and still be right. Two antiderivatives of the same function differ by a constant, and trig answers can be written in several equal forms.
• ln|x| is shown with bars when x could be negative. When the inside can never be negative, such as x^2 + 1, the bars are dropped.
• Write 1/(2x) with brackets. Typing 1/2x means (1/2) times x.

To check an answer, differentiate it with the derivative calculator. To see the curve, open the graphing calculator. For equations, there is the algebra calculator.