Free Online Calculator

Integral Calculator

Name: Integral Calculator
Author: AskMathAI

Calculate integrals step-by-step with our advanced integration calculator. Perfect for finding $\int x^2 dx$ or $\int_0^1 x^2 dx$ with detailed solutions.

Integral Calculator

Enter a function and get step-by-step integral solutions

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Master Integration with Our Advanced Calculator

Our integral calculator is designed to help students, teachers, and professionals solve 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 integrals.

The integration calculator handles both definite and indefinite integrals: polynomials, trigonometric functions, exponential and logarithmic functions, and composite functions using various integration techniques. It can find $\int x^2 dx = \frac{x^3}{3} + C$ , $\int \sin(x) dx = -\cos(x) + C$ , and complex integrals like $\int e^{x^2} \cdot x dx$ . Our calculus calculator is particularly useful for university calculus courses and engineering applications, where integral calculations are essential.

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

Types of Integrals

Function Type	Example	Integral	Difficulty Level
Polynomial	$\int x^2 dx$	$\frac{x^3}{3} + C$	Beginner
Trigonometric	$\int \sin(x) dx$	$-\cos(x) + C$	Intermediate
Exponential	$\int e^x dx$	$e^x + C$	Intermediate
Logarithmic	$\int \frac{1}{x} dx$	$\ln\|x\| + C$	Intermediate
Definite Integral	$\int_0^1 x^2 dx$	$\frac{1}{3}$	Advanced

Common Mistakes to Avoid

Forgetting the Constant

When calculating indefinite integrals, always add the constant of integration $C$ . For example: $\int x^2 dx = \frac{x^3}{3} + C$ , not just $\frac{x^3}{3}$ .

Wrong Integration Technique

For $\int xe^x dx$ , use integration by parts, not substitution. The choice of integration technique is crucial for success.

Sign Errors

Remember that $\int \sin(x) dx = -\cos(x) + C$ and $\int \cos(x) dx = \sin(x) + C$ . These negative signs are crucial for correct results.

How to Calculate Integrals

Integration is the reverse process of differentiation. It finds the antiderivative of a function, which represents the area under the curve of the original function.

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

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

Integration Techniques

Power Rule

For $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (n ≠ -1)

Substitution

Use $u$ -substitution for composite functions

Integration by Parts

Use $\int u dv = uv - \int v du$

Partial Fractions

Decompose rational functions

Common Integrals

$\int \sin(x) dx = -\cos(x) + C$

$\int e^x dx = e^x + C$

Examples

Polynomial Function

$\int x^2 dx$

Solution:

Apply power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
Substitute n = 2: $\frac{x^{2+1}}{2+1} + C$
Simplify: $\frac{x^3}{3} + C$

$\frac{x^3}{3} + C$

Trigonometric Function

$\int \sin(x) dx$

Solution:

Use trigonometric integral rule
$\int \sin(x) dx = -\cos(x) + C$

$-\cos(x) + C$

Definite Integral

$\int_0^1 x^2 dx$

Solution:

Find antiderivative: $\frac{x^3}{3}$
Apply limits: $\frac{1^3}{3} - \frac{0^3}{3}$
Calculate: $\frac{1}{3} - 0 = \frac{1}{3}$

$\frac{1}{3}$

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Frequently Asked Questions

What is an integral?

An integral is the reverse process of differentiation. It finds the antiderivative of a function and represents the area under the curve of the original function. There are two types: indefinite integrals (with constant C) and definite integrals (with limits).

What is the difference between definite and indefinite integrals?

An indefinite integral $\int f(x) dx$ gives the antiderivative $F(x) + C$ with a constant of integration. A definite integral $\int_a^b f(x) dx$ gives a numerical value representing the area under the curve from x = a to x = b.

How do I choose the right integration technique?

Start with basic rules (power rule, trigonometric integrals). For composite functions, try substitution. For products of functions, use integration by parts. For rational functions, consider partial fractions. Practice helps develop intuition for choosing the right method.

Can this calculator handle complex functions?

Yes, our integral calculator can handle polynomials, trigonometric functions, exponential and logarithmic functions, and many composite functions using various integration techniques.

Is this calculator free to use?

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

How accurate are the solutions?

Our calculator provides exact symbolic solutions and shows step-by-step work to help you understand the process. It follows standard mathematical conventions and integration rules.

Integral Calculator