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Antiderivative Calculator

Name: Antiderivative Calculator
Author: AskMathAI

Calculate antiderivatives step-by-step with our advanced integration calculator. Perfect for finding $\int f(x) \, dx$ or indefinite integrals with detailed explanations.

Antiderivative Calculator

Enter a function and get its antiderivative with step-by-step solutions

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Master Antiderivatives with Our Advanced Calculator

Our antiderivative calculator is designed to help students, teachers, and professionals solve indefinite integration problems efficiently. Whether you're working on calculus homework, preparing foruniversity mathematics exams, or tackling engineering problems, this tool provides comprehensive step-by-step solutions that enhance your understanding of integration.

The antiderivative $\int f(x) \, dx$ represents the family of functions whose derivative is $f(x)$ . It's the reverse process of differentiation and includes an arbitrary constant $C$ . Our calculus calculator is particularly useful for university calculus courses and engineering applications, where antiderivatives are used to solve differential equations, calculate areas, and model physical phenomena.

Perfect for high school calculus students learning integration, university studentsin advanced calculus courses, engineering students applying antiderivatives to real-world problems, andprofessionals who need quick mathematical solutions. The antiderivative calculator provides not just answers, but detailed explanations that help you understand the underlying concepts and improve your mathematical skills.

Types of Antiderivatives

Function Type	Example	Antiderivative	Difficulty Level
Power Functions	$x^n$	$\frac{x^{n+1}}{n+1} + C$ (n ≠ -1)	Beginner
Trigonometric	$\sin(x)$	$-\cos(x) + C$	Beginner
Exponential	$e^x$	$e^x + C$	Beginner
Logarithmic	$\frac{1}{x}$	$\ln\|x\| + C$	Intermediate
Rational Functions	$\frac{1}{x^2+1}$	$\arctan(x) + C$	Intermediate

Common Mistakes to Avoid

Forgetting the Constant

Always include the arbitrary constant $C$ in indefinite integrals. For example: $\int x^2 \, dx = \frac{x^3}{3} + C$ , not just $\frac{x^3}{3}$ .

Power Rule for n = -1

The power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ doesn't work for $n = -1$ . Instead, $\int \frac{1}{x} \, dx = \ln|x| + C$ .

Chain Rule Confusion

For $\int f(ax+b) \, dx$ , use substitution: $u = ax+b$ . Don't forget to divide by $a$ : $\int f(ax+b) \, dx = \frac{1}{a}F(ax+b) + C$ .

How to Calculate Antiderivatives

An antiderivative is a function whose derivative equals the original function. It's calculated using various integration techniques.

$\int f(x) \, dx = F(x) + C$

Where $F(x)$ is the antiderivative of $f(x)$ and $C$ is the arbitrary constant of integration.

Integration Techniques

Power Rule

For x^n: ∫x^n dx = x^(n+1)/(n+1) + C

Trigonometric

Use standard trig antiderivatives

Exponential

∫e^x dx = e^x + C

Substitution

Use u-substitution for complex functions

Important Integration Rules

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

$\int \frac{1}{x} \, dx = \ln|x| + C$

Examples

Power Function

$\int x^3 \, dx$

Solution:

Apply power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$
For n = 3: $\int x^3 \, dx = \frac{x^4}{4} + C$

$\frac{x^4}{4} + C$

Trigonometric

$\int \cos(x) \, dx$

Solution:

Use standard trig antiderivative: $\int \cos(x) \, dx = \sin(x) + C$

$\sin(x) + C$

Exponential

$\int e^{2x} \, dx$

Solution:

Use substitution: u = 2x, du = 2dx
$\int e^{2x} \, dx = \frac{1}{2} \int e^u \, du = \frac{1}{2} e^u + C$
Substitute back: $\frac{1}{2} e^{2x} + C$

$\frac{1}{2} e^{2x} + C$

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Frequently Asked Questions

What is an antiderivative?

An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative equals $f(x)$ . In other words, $F'(x) = f(x)$ . The antiderivative is also called the indefinite integral and includes an arbitrary constant $C$ .

What is the difference between antiderivative and definite integral?

An antiderivative (indefinite integral) gives a family of functions with an arbitrary constant $C$ , while a definite integral gives a specific numerical value. For example, $\int x^2 \, dx = \frac{x^3}{3} + C$ vs $\int_0^1 x^2 \, dx = \frac{1}{3}$ .

Why do we add +C to antiderivatives?

We add $+C$ because the derivative of any constant is zero. So if $F(x)$ is an antiderivative of $f(x)$ , then $F(x) + C$ is also an antiderivative for any constant $C$ . This represents the family of all possible antiderivatives.

How do I find the antiderivative of a polynomial?

For a polynomial, use the power rule for each term: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ ). For example, $\int (x^2 + 3x + 2) \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + 2x + C$ .

Can this calculator handle complex functions?

Yes, our antiderivative calculator can handle polynomials, trigonometric functions, exponential functions, logarithmic functions, and rational functions. It provides step-by-step solutions for various types of antiderivatives.

Is this calculator free to use?

Yes, our antiderivative calculator is completely free to use with no limitations. You can calculate as many antiderivatives as you need.

Antiderivative Calculator