Antiderivative Calculator – Free Tool with Steps and Answers

Q: 4. What is the antiderivative of x2?

The antiderivative of x2 is (x3)/3 + C, where C is the constant of integration. This is found by applying the power rule of integration.

Q: 6. What is the antiderivative of ex?

The antiderivative of ex is simply ex + C.

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How to Find the Antiderivative of Any Function Online

Antiderivative Calculator

What is Antiderivative Calculator?

The Antiderivative Calculator is an easy-to-use online tool that helps you find the antiderivative (also known as the indefinite integral or primitive function) of a given mathematical function. In calculus, the antiderivative of a function f(x) is a function F(x) such that the derivative of F(x) is f(x), i.e., F'(x) = f(x). This calculator instantly provides the antiderivative along with the constant of integration for polynomials, trigonometric, exponential, logarithmic, and other common functions.
Antiderivatives are widely used in mathematics and science to reconstruct original functions from their rates of change. By using this calculator, students and professionals can solve calculus homework, check their work, or learn integration steps quickly.

Formula or Logic Behind Antiderivative Calculator

The core logic for the antiderivative calculator is built upon the reverse process of differentiation. If f(x) is a function, an antiderivative F(x) satisfies F'(x) = f(x). The collection of all antiderivatives of f(x) is called the indefinite integral and is written as:

∫ f(x) dx = F(x) + C

Here, C is the constant of integration (because the derivative of any constant is zero). Common formulas used include:
- ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1
- ∫ sin(x) dx = -cos(x) + C
- ∫ cos(x) dx = sin(x) + C
- ∫ eˣ dx = eˣ + C
- ∫ 1/x dx = ln|x| + C

Popular Antiderivatives Table

Function f(x)	Antiderivative F(x) + C
x	x²/2 + C
xⁿ (n ≠ -1)	xⁿ⁺¹/(n+1) + C
sin(x)	-cos(x) + C
cos(x)	sin(x) + C
e^x	e^x + C
1/x	ln\|x\| + C
ln(x)	x ln x - x + C
tan(x)	-ln\|cos(x)\| + C

Steps to Use the Antiderivative Calculator

Enter the required function f(x) (for example: x^2, sin(x), e^x)
Select the variable of integration (x or y)
Click on the 'Calculate' button
Get instant results with the calculated antiderivative and steps

Why Use Vedantu’s Antiderivative Calculator?

Vedantu’s Antiderivative Calculator is designed by expert educators and tailored for school, college, and competitive exam needs. It is mobile-friendly, gives instant, accurate answers with step-by-step logic, and covers almost all popular integration formulas. Trusted by thousands of Indian students and recommended by teachers, it helps you master Calculus in less time and check your work efficiently. Explore Integral Calculus concepts.

Real-life Applications of Antiderivative Calculator

Understanding antiderivatives is crucial in many academic and professional fields. For example:

Physics: Calculating displacement from velocity, or total quantity from rate of change
Engineering: Measuring area under curves for signal processing or construction
Economics: Computing accumulated investment growth from a rate function
Biology: Modeling population growth and decay using integral equations
Academics: Solving calculus problems in school and university coursework

For example, a student learning integration can practice problems and verify answers using Vedantu’s calculator. Professionals may use it to double check calculations in research or work. It is also useful for self-study and exam revisions. Read more on Integration here. Brush up Derivatives and Limits here. See Applications of Integrals in real-world scenarios.

All solutions and calculators on Vedantu are reviewed by expert Maths educators and are fully aligned with NCERT and CBSE standards.

FAQs on Antiderivative Calculator – Free Tool with Steps and Answers

1. What is an antiderivative?

An antiderivative, also known as an indefinite integral or primitive function, is the reverse process of differentiation. It's a function whose derivative is the original function. In simpler terms, if you know the rate of change (derivative), the antiderivative helps find the original function.

2. What is the difference between an antiderivative and a definite integral?

The key difference lies in the result. An antiderivative gives a family of functions (because the constant of integration 'C' can be any value), while a definite integral provides a numerical value representing the area under a curve between two specific points. The definite integral is calculated using the antiderivative.

3. How do I find the antiderivative of a function?

Finding the antiderivative involves applying integration rules, which are essentially the reverse of differentiation rules. For example, the antiderivative of xⁿ is xⁿ⁺¹/(n+1) + C, where C is the constant of integration. There are specific rules for functions like trigonometric functions (sin x, cos x), exponential functions (e^x), and logarithmic functions (ln x).

4. What is the antiderivative of x²?

The antiderivative of x² is (x³)/3 + C, where C is the constant of integration. This is found by applying the power rule of integration.

5. What is the antiderivative of sin(x)?

The antiderivative of sin(x) is -cos(x) + C. Remember the constant of integration!

6. What is the antiderivative of e^x?

The antiderivative of e^x is simply e^x + C.

7. What is the antiderivative of 1/x?

The antiderivative of 1/x is ln|x| + C. The absolute value ensures the argument of the natural logarithm is positive.

8. What are some real-world applications of antiderivatives?

Antiderivatives have many applications: Calculating displacement from velocity in physics, finding the area under a curve (used in various fields), determining volume, modeling population growth, and more. Essentially, whenever you need to find a function from its rate of change, antiderivatives are crucial.

9. How do I use an online antiderivative calculator?

Typically, you'll enter the function you want to find the antiderivative of into the calculator's input field. Then, click the 'Calculate' button. The calculator will return the antiderivative, often including the steps involved in the calculation. It's a quick and easy way to check your work or get help solving integration problems.

10. What is the constant of integration (C)?

The constant of integration (C) is added to the result of an indefinite integral because the derivative of a constant is zero. Therefore, many functions can have the same derivative. The constant C accounts for this ambiguity; it represents an arbitrary constant that can take any real value.

11. What are the basic rules of antidifferentiation?

The basic rules mirror the rules of differentiation, but in reverse. Key rules include the power rule (adding 1 to the exponent and dividing by the new exponent), the sum/difference rule (integrating term by term), and rules for trigonometric, exponential, and logarithmic functions. Understanding these rules is fundamental to finding antiderivatives.