For \(\int x e^x\,dx\), the calculator picks integration by parts. It explains why u = x is the best choice (the LIATE rule), and then checks the answer by taking its derivative.

Try this example → Learn integration by parts

Choose u and dv

\(u = x,\quad dv = e^x\,dx\)

Find du and v

\(du = dx,\quad v = e^x\)

Use the formula

\(\int u\,dv = uv – \int v\,du\)

Answer

\(x e^x – e^x + C = e^x(x-1) + C\)

✓ Checked

\(\frac{d}{dx}\big[e^x(x-1)\big] = x e^x\)

Why students use it

Built to help you learn, not just copy

A good calculator should teach you the method. Here is what you get with every problem.

Steps in plain English

Every step says what rule was used and why, in simple words.

Answers that are checked

We take the derivative of each answer. If it does not match, you never see it.

Graphs you can read

See your function and its antiderivative on the same graph.

Math keyboard

Tap buttons for √, π, sin, ln and powers. Great on a phone.

Share links

Send a problem to a friend or teacher with one link.

Works on any device

Fast on phones, tablets and laptops. No app and no sign-up.

Learn the methods

Short, clear guides with worked examples for each rule.

Power rule U-substitution Integration by parts Trig functions Exponential functions Antiderivative rules Constant of integration Practice problems

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Meet the author

Prof. Adrian Cole

Mathematics educator · Calculus and integration

Prof. Adrian Cole makes derivatives, antiderivatives and integration easier to understand. He writes every guide on this site and tests every calculator against known problems. His lessons show not only how to solve a problem, but why the method works.

About us How we check the math

What does an antiderivative calculator do?

An antiderivative calculator finds a function \(F(x)\) whose derivative is your function \(f(x)\), then adds the constant \(C\). For example, the antiderivative of \(2x\) is \(x^2 + C\). This calculator also shows each integration step and checks the result by taking its derivative.

How to Use the Antiderivative Calculator

Type your function in the box. You can write 3x^2 without a star.
Look at the preview under the box. It shows how your input was read, as real math.
Choose the variable if it is not x.
Press Solve or Enter. Open the steps, the graph, or press Share link to send the problem to someone.

Input Guide

Math	Type	Math	Type
\(x^3\)	`x^3`	\(\sqrt{x}\)	`sqrt(x)`
\(5x\)	`5x`	\(\sqrt[3]{x}\)	`x^(1/3)`
\(\frac{1}{x}\)	`1/x`	\(\ln x\)	`ln(x)`
\(e^{-x}\)	`e^(-x)`	\(\pi\)	`pi`
\(\cos^2 x\)	`cos(x)^2`	\(\sec x\tan x\)	`sec(x)tan(x)`

What This Antiderivative Calculator Can Do

You may know this tool by another name: an antiderivative solver, an anti-differentiation calculator, an integral finder or a “reverse derivative” calculator. They all do the same job: given \(f(x)\), find a function \(F(x)\) whose derivative is \(f(x)\).

General antiderivative with steps. Every answer includes \(+C\), so you get the whole family of antiderivatives, not just one. Each step names the rule used.
A particular antiderivative. If your problem gives a starting value such as \(F(0) = 2\), find \(F(x) + C\) here, then solve for \(C\) as shown in the constant of integration guide.
A second antiderivative. Integrate the answer again (drop the \(+C\) first, then add a new one at the end). For example, the second antiderivative of \(6x\) is \(x^3 + C_1x + C_2\).
Any variable. Change the variable box to \(t\), \(y\) or \(\theta\) for physics and engineering problems.

Handheld calculators such as the TI-84 can only give a definite integral as a number. To get an antiderivative formula, you need a computer algebra system like this one. See which calculators can do integrals.

What Is an Antiderivative?

An antiderivative of \(f(x)\) is a function \(F(x)\) whose derivative is \(f(x)\). In symbols, \(F'(x) = f(x)\).

For example, the derivative of \(x^2\) is \(2x\). So \(x^2\) is an antiderivative of \(2x\). But so are \(x^2 + 1\) and \(x^2 – 7\), because constants disappear when you take a derivative. That is why we add + C:

\[\int 2x\,dx = x^2 + C.\]

New to the topic? Start with What Is an Antiderivative? A Simple Guide for Beginners and why the constant of integration matters.

Antiderivative Formulas

These are the formulas used most often. The calculator applies them for you and tells you which one it used.

Constant\[\int k\,dx = kx + C\]

Power rule\[\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\]for any \(n \ne -1\)

One over x\[\int \dfrac{1}{x}\,dx = \ln|x| + C\]

Exponential\[\int e^x\,dx = e^x + C\]

Any base\[\int a^x\,dx = \dfrac{a^x}{\ln a} + C\]

Sine\[\int \sin x\,dx = -\cos x + C\]

Cosine\[\int \cos x\,dx = \sin x + C\]

Secant squared\[\int \sec^2 x\,dx = \tan x + C\]

Tangent\[\int \tan x\,dx = -\ln|\cos x| + C\]

Natural log\[\int \ln x\,dx = x\ln x – x + C\]

Arctangent\[\int \dfrac{1}{1+x^2}\,dx = \tan^{-1} x + C\]

Arcsine\[\int \dfrac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1} x + C\]

Learn each rule with examples in Antiderivative Rules You Must Know. There are also full guides for sin(x), ln(x), eˣ, and 1/x.

Worked Examples

Example 1: \(\int (6x^2 – 4x + 1)\,dx\)

Use the power rule on each term: \(\frac{6x^3}{3} – \frac{4x^2}{2} + x = 2x^3 – 2x^2 + x\).

Answer: \(2x^3 – 2x^2 + x + C\).

Example 2: \(\int \frac{3}{x}\,dx\)

Pull out the 3, then use \(\int\frac1x\,dx = \ln|x|\).

Answer: \(3\ln|x| + C\).

Example 3: \(\int \cos(4x)\,dx\)

The inside is \(4x\), so divide by 4: \(\frac14\sin(4x)\).

Answer: \(\frac14\sin(4x) + C\).

Example 4: \(\int x\cos(x^2)\,dx\)

Use u-substitution with \(u = x^2\), \(du = 2x\,dx\).

Answer: \(\frac12\sin(x^2) + C\).

How to Check Any Antiderivative

Take the derivative of your answer. If you get back the function you started with, your answer is right. The calculator does this for every problem and shows it under “Check by differentiating.” Read more in how to check your antiderivative answer.

Common Mistakes

Forgetting + C.
Writing \(\int \sin x\,dx = \cos x\). The correct answer is \(-\cos x\).
Using the power rule on \(x^{-1}\). Use \(\ln|x|\) instead.
Splitting a product into two integrals. \(\int x e^x\,dx \ne \int x\,dx\cdot\int e^x\,dx\).

See all five big ones in Top 5 Mistakes Students Make When Solving Antiderivatives.

Frequently Asked Questions

Is an antiderivative the same as an integral?

An antiderivative is the same as an indefinite integral. A definite integral is a number found from an antiderivative. Use the definite integral calculator for those.

What is anti-differentiation?

Anti-differentiation is the process of finding an antiderivative: working backward from a derivative to the original function. It is the same thing as finding an indefinite integral, and the answer always includes \(+C\).

How do I find an antiderivative on a TI-84?

You can’t get a formula on a TI-84. It only computes definite integrals as numbers with fnInt. For the antiderivative itself, use this calculator or a CAS calculator such as the TI-89 or TI-Nspire CX CAS.

Can every function be integrated?

Every continuous function has an antiderivative, but some cannot be written with normal functions. Examples are \(e^{x^2}\) and \(\frac{\sin x}{x}\). The calculator tells you when this happens.

Why does my textbook answer look different?

Two answers can look different and both be right. They may differ by a constant, or use a trig identity. For example, \(-\ln|\cos x|\) and \(\ln|\sec x|\) are equal.

Does this calculator work on phones?

Yes. Tap Math keyboard to get buttons for √, π, sin, ln, and more.

Who made this calculator?

It was built and math-checked by Prof. Adrian Cole. Read more about us, or contact us if you find a problem.

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