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Antiderivative Calculator

Enter a function and get a function whose derivative it is, with the constant of integration.

Kind of integral
3x2+2x+53x^{2}+2x+53x2+2x+5

Antiderivative

∫3x2+2x+5 dx=x3+x2+5x+C\int 3x^{2}+2x+5\,dx=x^{3}+x^{2}+5x+C∫3x2+2x+5dx=x3+x2+5x+C
  • f(x)
  • F(x), with C = 0
Table of values
Graph of 3*x^2+2*x+5: sampled values across the visible range
xf(x)F(x), with C = 0
-6101-210
-4.864.52-111.55
-3.636.68-51.696
-2.417.48-20.064
-1.26.92-6.288
050
1.211.729.168
2.427.0831.584
3.651.0877.616
4.883.72157.63
6125282

Step-by-step working

  1. 1Linearity
    ∫3x2+2x+5 dx\int 3x^{2}+2x+5\,dx∫3x2+2x+5dx
    ↓
    ∫3x2 dx+∫2x dx+∫5 dx\int 3x^{2}\,dx+\int 2x\,dx+\int 5\,dx∫3x2dx+∫2xdx+∫5dx

    Integrate each term separately.

  2. 2Constant multiple rule
    ∫3x2 dx\int 3x^{2}\,dx∫3x2dx
    ↓
    3⋅∫x2 dx3\cdot \int x^{2}\,dx3⋅∫x2dx
  3. 3Power rule
    ∫x2 dx\int x^{2}\,dx∫x2dx
    ↓
    x33\frac{x^{3}}{3}3x3​

    x^n integrates to x^(n+1) / (n + 1) for n other than -1.

  4. 4Constant multiple rule
    ∫2x dx\int 2x\,dx∫2xdx
    ↓
    2⋅∫x dx2\cdot \int x\,dx2⋅∫xdx
  5. 5Power rule
    ∫x dx\int x\,dx∫xdx
    ↓
    x22\frac{x^{2}}{2}2x2​
  6. 6Constant rule
    ∫5 dx\int 5\,dx∫5dx
    ↓
    5x5x5x
  7. 7Add the constant
    x3+x2+5xx^{3}+x^{2}+5xx3+x2+5x
    ↓
    x3+x2+5x+Cx^{3}+x^{2}+5x+Cx3+x2+5x+C

    Every antiderivative differs by a constant.

Worked solution: antiderivative of 3x^2 + 2x + 5

Worked steps

  1. 1Linearity
    ∫3x2+2x+5 dx\int 3x^{2}+2x+5\,dx∫3x2+2x+5dx
    ↓
    ∫3x2 dx+∫2x dx+∫5 dx\int 3x^{2}\,dx+\int 2x\,dx+\int 5\,dx∫3x2dx+∫2xdx+∫5dx

    Integrate each term separately.

  2. 2Constant multiple rule
    ∫3x2 dx\int 3x^{2}\,dx∫3x2dx
    ↓
    3⋅∫x2 dx3\cdot \int x^{2}\,dx3⋅∫x2dx
  3. 3Power rule
    ∫x2 dx\int x^{2}\,dx∫x2dx
    ↓
    x33\frac{x^{3}}{3}3x3​

    x^n integrates to x^(n+1) / (n + 1) for n other than -1.

  4. 4Constant multiple rule
    ∫2x dx\int 2x\,dx∫2xdx
    ↓
    2⋅∫x dx2\cdot \int x\,dx2⋅∫xdx
  5. 5Power rule
    ∫x dx\int x\,dx∫xdx
    ↓
    x22\frac{x^{2}}{2}2x2​
  6. 6Constant rule
    ∫5 dx\int 5\,dx∫5dx
    ↓
    5x5x5x
  7. 7Add the constant
    x3+x2+5xx^{3}+x^{2}+5xx3+x2+5x
    ↓
    x3+x2+5x+Cx^{3}+x^{2}+5x+Cx3+x2+5x+C

    Every antiderivative differs by a constant.

Answer

∫3x2+2x+5 dx=x3+x2+5x+C\int 3x^{2}+2x+5\,dx=x^{3}+x^{2}+5x+C∫3x2+2x+5dx=x3+x2+5x+C
  1. Tools
  2. Mathematics
  3. Integral Calculator
  4. Antiderivative Calculator

About Antiderivative Calculator

An antiderivative of f is a function F whose derivative is f. This page opens with 3x^2 + 2x + 5, whose antiderivative is x^3 + x^2 + 5x + C. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

Each term is reversed with the power rule: raise the exponent by one and divide by the new exponent. The constant 5 becomes 5x. Every antiderivative of the same function differs only by a constant, which is why the answer ends with + C.

The graph shows the function together with the antiderivative drawn with C = 0. If no elementary rule applies the calculator says so instead of guessing.

How to use Antiderivative Calculator

  1. 1Type the function in the Function box, for example x*e^x, or use the key pad. The preview shows how it was read.
  2. 2Choose Indefinite for an antiderivative, or Definite to enter a lower and an upper bound.
  3. 3Write bounds as numbers or constant expressions such as pi/2. Type inf or -inf for an improper integral.
  4. 4Read the result card. A definite integral shows the exact value when one exists and a labelled numeric estimate otherwise.
  5. 5Check the graph: the function is drawn with its area shaded between the bounds, or with its antiderivative for an indefinite integral.
  6. 6Open the steps to see which method was used, and copy the answer with the copy button.

Features

  • Indefinite integrals with the constant of integration.
  • Definite integrals with exact values, including bounds written with pi and with inf or -inf.
  • Step-by-step working that names the rule: power rule, substitution, integration by parts, partial fractions and trigonometric identities.
  • A shaded area under the curve for definite integrals, on a graph you can zoom and drag.
  • A clear label when the result is a numeric estimate instead of an exact value.
  • An honest "no elementary rule found" message instead of a guess.
  • A live preview and a math key pad, so input is easy on a phone.
  • Checks improper integrals and says when one does not settle on a finite value.

Worked examples

  • Antiderivative of 3x^2 + 2x + 5 → gives x^3 + x^2 + 5x + C

    Power rule on each term.

Privacy: how it works

  • The integral is worked out by code running in your browser tab. Your functions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Differentiate your answer with the derivative calculator to check it. You should get the original function back.
  • For an area between a curve and the x axis, remember that parts below the axis count as negative in a definite integral.
  • An improper integral with a bound at infinity only has a value if the function dies away fast enough. 1/x^2 does, 1/x does not.
  • Use * between factors when you can, for example x*e^x, and write powers of e as e^x.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your functions are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your functions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What is the antiderivative of 3x^2 + 2x + 5?

It is x^3 + x^2 + 5x + C. Differentiating it gives 3x^2 + 2x + 5 again, which confirms the answer.

Why do I need + C?

Constants disappear when you differentiate, so many functions share the same derivative. The C stands for any constant.

Is an antiderivative the same as an integral?

The indefinite integral is the family of all antiderivatives, written with + C. A definite integral is a number, found by evaluating an antiderivative at two bounds.

More Integral Calculator pages

  • Integral CalculatorFind an antiderivative or the area between two bounds, with the method named at every step.
  • Integral of ln(x)Step-by-step integral of ln(x), with the method named and a graph of the function and its antiderivative.
  • Integral of sin^2(x)Step-by-step integral of sin^2(x), with the method named and a graph of the function and its antiderivative.
  • Integral of x*e^xStep-by-step integral of x*e^x, with the method named and a graph of the function and its antiderivative.
  • Integral of 1/xStep-by-step integral of 1/x, with the method named and a graph of the function and its antiderivative.
  • Integral of tan(x)Step-by-step integral of tan(x), with the method named and a graph of the function and its antiderivative.
  • Integral of sec(x)Step-by-step integral of sec(x), with the method named and a graph of the function and its antiderivative.
  • Integral of e^(-x^2)Why e^(-x^2) has no closed-form antiderivative, and its numeric value over any range.
  • Integral of x*sin(x)Step-by-step integral of x*sin(x), with the method named and a graph of the function and its antiderivative.
  • Integral of 1/(1+x^2)Step-by-step integral of 1/(1+x^2), with the method named and a graph of the function and its antiderivative.
  • Integral of sqrt(x)Step-by-step integral of sqrt(x), with the method named and a graph of the function and its antiderivative.
  • Integral of cos^2(x)Step-by-step integral of cos^2(x), with the method named and a graph of the function and its antiderivative.
  • Integral of 1/(x^2-1)Step-by-step integral of 1/(x^2-1), with the method named and a graph of the function and its antiderivative.

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