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This free integral calculator solves indefinite and definite integrals and shows every step. Type your function, press Solve, and you get the answer, the rule used at each step, and a graph. Each answer is checked by taking its derivative, so you can trust the result.
Use the Indefinite tab to find an antiderivative (the answer ends in + C). Use the Definite tab when you have limits, like from 0 to 2, and want a number.
How to Use the Integral Calculator
- Type the function you want to integrate. For example:
x^2 sin(x). - Pick the variable. Most problems use x, but you can choose t, u, y, or θ.
- Choose Indefinite or Definite. For a definite integral, enter the lower and upper limits. You can type
piorinf. - Press Solve (or hit Enter). Read the answer, then open the steps to see how it was done.
How to Type Math
| You want | Type this |
|---|---|
| \(x^2\) | x^2 |
| \(3x\) | 3x or 3*x |
| \(\sqrt{x}\) | sqrt(x) |
| \(\ln x\) | ln(x) |
| \(e^{2x}\) | e^(2x) |
| \(\sin^2 x\) | sin(x)^2 or sin^2(x) |
| \(\tan^{-1} x\) | arctan(x) |
| \(|x|\) | abs(x) or |x| |
Under the input box you will see “We read this as”. It shows how the calculator understood your typing. If it looks wrong, add brackets.
What Is an Integral?
An integral “adds up” tiny pieces. There are two kinds:
- Indefinite integral: a family of functions whose derivative is your function. We write \(\int f(x)\,dx = F(x) + C\). The constant of integration C is there because the derivative of any constant is 0.
- Definite integral: a single number. It gives the signed area between the graph and the x-axis from \(x=a\) to \(x=b\): \(\int_a^b f(x)\,dx = F(b) – F(a)\).
Integration is the reverse of differentiation. If you want to learn the basics first, read what an antiderivative is and the difference between definite and indefinite integrals.
Integration Methods This Calculator Uses
The calculator picks the method a teacher would pick, then shows it to you step by step.
| Method | When it is used | Example |
|---|---|---|
| Power rule | Powers of x | \(\int x^4\,dx = \frac{x^5}{5} + C\) |
| Standard integrals | sin, cos, ex, 1/x and more | \(\int \cos x\,dx = \sin x + C\) |
| U-substitution | A function and its inside derivative | \(\int 2x\cos(x^2)\,dx\) |
| Integration by parts | Products like x·ex or x·ln x | \(\int x e^x\,dx\) |
| Partial fractions | Fractions of polynomials | \(\int \frac{1}{x^2-5x+6}\,dx\) |
| Trig identities | Powers of sin and cos | \(\int \sin^2 x\,dx\) |
Worked Examples
Example 1: \(\int (3x^2 + 4x – 5)\,dx\)
Integrate each term with the power rule. \(\int 3x^2\,dx = x^3\), \(\int 4x\,dx = 2x^2\), and \(\int -5\,dx = -5x\).
Answer: \(x^3 + 2x^2 – 5x + C\).
Check: The derivative of \(x^3 + 2x^2 – 5x\) is \(3x^2 + 4x – 5\). ✓
Example 2: \(\int x e^x\,dx\)
This is a product, so use integration by parts. Let \(u = x\) and \(dv = e^x\,dx\). Then \(du = dx\) and \(v = e^x\).
\(\int x e^x\,dx = x e^x – \int e^x\,dx = x e^x – e^x + C\).
Answer: \(e^x(x – 1) + C\).
Example 3: \(\int_0^{\pi} \sin x\,dx\)
An antiderivative of \(\sin x\) is \(-\cos x\). So the value is \(-\cos(\pi) – (-\cos 0) = 1 + 1 = 2\).
Answer: 2. The area under one hump of the sine wave is exactly 2.
Common Mistakes to Avoid
- Forgetting + C. Every indefinite integral needs it.
- Using the power rule on 1/x. \(\int x^{-1}\,dx\) is \(\ln|x| + C\), not \(x^0/0\). See the antiderivative of 1/x.
- Integrating a product term by term. \(\int x\cos x\,dx\) is not \(\frac{x^2}{2}\sin x\). Products need parts or substitution.
- Dropping the chain factor. \(\int \cos(3x)\,dx = \frac{1}{3}\sin(3x) + C\). Don’t forget the \(\frac13\).
We explain these in more detail in the top mistakes students make with antiderivatives.
Why Some Integrals Have No Formula
A few functions, like \(e^{x^2}\) or \(\frac{\sin x}{x}\), do not have an antiderivative made of normal functions. No calculator can give a simple formula for them. In that case this tool tells you so. You can still find a definite integral of these functions as a decimal with the definite integral calculator.
Learn the Math Behind the Integral Calculator
Integrals add up change. Use the indefinite form to find a formula with + C, or the definite form to get a number such as an area.
- GuideDefinite vs indefinite integrals, explained
- GuideHow to find an antiderivative: an 8-question checklist
- GuideWhy integration is harder than differentiation
- Key termsIntegral, Indefinite integral, Definite integral
- Next toolsAntiderivative Calculator, Definite Integral Calculator, U-Substitution Calculator, Integration by Parts Calculator
Frequently Asked Questions
Is this integral calculator free?
Yes. It is free, there is no sign-up, and there is no limit on how many problems you solve.
Does it show steps?
Yes. Every answer comes with a step-by-step solution. Each step names the rule, like the power rule or u-substitution, and shows the math.
How do I know the answer is right?
The calculator differentiates its own answer and compares it with your function. When they match, you see a green ✓ Verified badge. You can do the same check by hand.
What is the difference between this and the antiderivative calculator?
They use the same math engine. The antiderivative calculator focuses on indefinite integrals. This page also handles definite integrals with limits.
Can I share a problem with a friend or teacher?
Yes. Press Share link. The link opens this page with your problem already solved.
Keep Learning
- How to find the antiderivative of any function
- Antiderivative rules you must know
- Antiderivatives of trigonometric functions
- 10 practice problems with full solutions