Free Antiderivative Calculator – Solve Any Integral Instantly

Welcome to our free antiderivative calculator — the fastest way to find the antiderivative of any function online. Whether you are a high school student just starting calculus, a college student preparing for an exam, or someone who needs a quick answer, this tool gives you instant results with a step-by-step explanation.

Just type your function in the box above, choose your variable, and hit Solve. You will see the answer in a clean, easy-to-read format in seconds.

How to Use This Antiderivative Calculator

Using the calculator is simple. Here is all you need to do:

1. Type your function in the input box. For example: x^2 + sin(x)
2. Select the variable you want to integrate with respect to (usually x).
3. Click the Solve button or press Enter on your keyboard.
4. Read the result and the step-by-step explanation below it.

💡 Input Syntax GuideMultiplication  →  use *   Example: 3*x  (not 3x)Powers          →  use ^   Example: x^2  (not x²)Functions       →  use ()  Example: sin(x), cos(x), ln(x)Natural log     →  ln(x)  or  log(x)Euler number    →  e  or  exp(x)Pi              →  piSquare root     →  sqrt(x)

You can also tap the Scientific Keyboard button to open a full keyboard with all math symbols. This is especially useful if you are on a mobile phone or tablet.

What Is an Antiderivative?

An antiderivative (also called an indefinite integral) is the reverse of a derivative. If you know how a function changes, the antiderivative tells you the original function.

Here is the simplest way to think about it:

• The derivative of x² is 2x.
• So the antiderivative of 2x is x² + C.
• The “+ C” is the constant of integration — it represents any constant number that disappears when you differentiate.

In math notation, we write the antiderivative using the integral symbol ∫. For example:
  ∫ 2x dx  =  x² + C

Finding antiderivatives is one of the most important skills in calculus. It is used to calculate area under a curve, total distance traveled, accumulated quantities, and much more.

The Most Important Antiderivative Rules

Before you start solving, it helps to know the key rules. Our calculator applies these rules automatically — but understanding them will make you a better math student.

1. The Power Rule

This is the most used rule in integration. For any number n (except n = −1):

   ∫ xⁿ dx  =  xⁿ⁺¹ / (n+1)  +  C

Example: ∫ x³ dx = x⁴/4 + C

2. The Constant Rule

The antiderivative of any constant k is k times x:

   ∫ k dx  =  kx  +  C

Example: ∫ 5 dx = 5x + C

3. The Sum and Difference Rule

You can split a sum or difference into separate integrals:

   ∫ [f(x) ± g(x)] dx  =  ∫ f(x) dx  ±  ∫ g(x) dx

4. The Constant Multiple Rule

A constant multiplied by a function can be pulled outside the integral:

   ∫ k·f(x) dx  =  k · ∫ f(x) dx

Antiderivative Formula Reference Table

Here is a quick reference for the most common antiderivative formulas. Bookmark this page so you always have it handy.

Function f(x)	Antiderivative F(x) + C	Rule Name
xⁿ  (n ≠ −1)	xⁿ⁺¹ / (n+1) + C	Power Rule
1/x  or  x⁻¹	ln|x| + C	Log Rule
eˣ	eˣ + C	Exp Rule
aˣ  (a > 0)	aˣ / ln(a) + C	Exp Rule
sin(x)	−cos(x) + C	Trig
cos(x)	sin(x) + C	Trig
tan(x)	−ln|cos(x)| + C	Trig
sec²(x)	tan(x) + C	Trig
csc²(x)	−cot(x) + C	Trig
sec(x)·tan(x)	sec(x) + C	Trig
ln(x)	x·ln(x) − x + C	IBP
√x  (sqrt(x))	(2/3)·x^(3/2) + C	Power Rule
1/(1+x²)	arctan(x) + C	Inv Trig
1/√(1−x²)	arcsin(x) + C	Inv Trig

Worked Examples – Step by Step

Let us walk through a few examples so you can see exactly how the antiderivative is found.

Example 1: Find the antiderivative of x³

Problem: ∫ x³ dx Step 1: Identify the rule. The function is x³ — this is a power function. Use the Power Rule.Step 2: Apply the Power Rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + CStep 3: Here n = 3. Add 1 to the exponent: 3 + 1 = 4.Step 4: Divide by the new exponent: x⁴ / 4. Answer: x⁴/4 + C

Example 2: Find the antiderivative of 3x² + 2x − 5

Problem: ∫ (3x² + 2x − 5) dx Step 1: Use the Sum Rule — integrate each term separately.Step 2: ∫ 3x² dx = 3 · x³/3 = x³Step 3: ∫ 2x dx = 2 · x²/2 = x²Step 4: ∫ −5 dx = −5xStep 5: Combine all terms and add + C. Answer: x³ + x² − 5x + C

Example 3: Find the antiderivative of sin(x)

Problem: ∫ sin(x) dx Step 1: Recognize this as a standard trigonometric integral.Step 2: From the trig formula table: ∫ sin(x) dx = −cos(x) + CReason: The derivative of −cos(x) is sin(x). So by reversing that, the antiderivative of sin(x) is −cos(x). Answer: −cos(x) + C

Why Do We Always Write + C?

You may have noticed that every antiderivative ends with + C. This is called the constant of integration, and it is always required for indefinite integrals.

Here is why. When you differentiate any constant number, the answer is always zero:

   d/dx (x² + 3)  =  2x        (the 3 disappears)

   d/dx (x² + 100) =  2x        (the 100 disappears)

This means the antiderivative of 2x could be x² + 3, or x² + 100, or x² + anything. Since we do not know which constant was there, we write + C to represent all possibilities.

💡 Remember+ C is NOT optional — always include it for indefinite integrals.C represents an unknown constant that vanishes during differentiation.For definite integrals (with limits), the C cancels out and is not needed.

Antiderivative vs Integral — What Is the Difference?

Many students use these two words interchangeably, but there is a small difference worth knowing.

• An antiderivative is the general function F(x) such that F'(x) = f(x). It always includes + C.
• An indefinite integral is written using the ∫ symbol and gives the same result as an antiderivative.
• A definite integral has upper and lower limits (like ∫₀¹) and gives a specific number — the area under the curve.

In practice, when people say “find the integral” or “find the antiderivative”, they usually mean the same thing — find the indefinite integral and add + C.

Real World Uses of Antiderivatives

Antiderivatives are not just a textbook topic. They show up everywhere in the real world:

• Velocity is the derivative of position. So to find position from velocity, you take the antiderivative.: Physics
• Electrical engineers use integration to analyze circuits and signal processing.: Engineering
• Economists find total cost from marginal cost by integrating.: Economics
• Doctors use integration to calculate how a drug accumulates in the bloodstream over time.: Medicine
• Rendering shadows, curves, and 3D shapes all rely on integration.: Computer Graphics

Frequently Asked Questions

What is the antiderivative of a constant?

The antiderivative of any constant k is k·x + C. For example, the antiderivative of 5 is 5x + C. You can verify this because the derivative of 5x is 5.

What is the antiderivative of 0?

The antiderivative of 0 is just C (a constant). This makes sense because the derivative of any constant is 0.

Can every function be integrated?

Not every function has a “nice” antiderivative that can be written in simple terms. For example, ∫ e^(x²) dx has no standard formula. However, most functions you meet in a calculus course can be integrated using the rules and techniques in this calculator.

What is the difference between integration and antiderivative?

They refer to the same process. “Integration” is the operation. “Antiderivative” is the result. Finding the antiderivative of f(x) is the same as evaluating the indefinite integral ∫ f(x) dx.

How do I check my antiderivative answer?

The easiest way to check is to differentiate your answer. If you get back the original function, your antiderivative is correct. For example, if you found that ∫ x² dx = x³/3 + C, take the derivative of x³/3: you get x². That confirms the answer is right.

Is this calculator free to use?

Yes, completely free. No sign-up, no login, no subscription. You can use it as many times as you want directly in your browser on any device — phone, tablet, or computer.

Tips for Getting the Best Results

• Always use * for multiplication — write 3*x, not 3x.
• Always use ^ for powers — write x^2, not x².
• Always use parentheses for function arguments — write sin(x), not sinx.
• For complex expressions, use parentheses to group terms — for example: (x+1)^2.
• Use the Scientific Keyboard if you are on mobile — it has all the symbols you need.
• Check your answer by differentiating the result mentally.

Start Solving Now

This antiderivative calculator was built to help students and learners like you get accurate answers quickly. Whether you are working on homework, studying for an exam, or just curious about calculus, you can use this tool for free, anytime.

Type your function in the calculator at the top of this page and click Solve. You will get the antiderivative with a full step-by-step explanation — in seconds.

💡 Quick SummaryAntiderivative = reverse of a derivativeAlways add + C for indefinite integralsUse the Power Rule for xⁿ functionsUse the Formula Table above for trig, log, and exponential functionsType your function above and click Solve for instant results