Integral Calculator
Calculate integrals step by step with our free online calculator. Compute both definite and indefinite integrals, view the solution steps, and visualize the results graphically. Perfect for calculus students and professionals.
How to Use the Integral Calculator
- Enter the function you want to integrate
- Specify the variable of integration (default is x)
- Choose between definite or indefinite integral
- For definite integrals, enter the lower and upper bounds
- Click "Calculate Integral" to see the solution
Solution
Enter a function and click "Calculate Integral" to see the solution.
The step-by-step solution will appear here after calculation.
A graphical representation will appear here for definite integrals.
About Integrals
Integrals are fundamental concepts in calculus that represent the area under a curve or the accumulation of quantities. There are two main types of integrals: definite and indefinite.
Indefinite Integrals
An indefinite integral represents the antiderivative of a function. It is written as:
\[ \int f(x)\,dx = F(x) + C \]
Where \( F(x) \) is the antiderivative of \( f(x) \), and \( C \) is the constant of integration.
Definite Integrals
A definite integral calculates the net area between the function and the x-axis over a specified interval [a, b]:
\[ \int_{a}^{b} f(x)\,dx \]
The result is a number representing this net area, which can be interpreted as accumulated change.
Common Integration Techniques
This calculator supports several integration methods:
- Basic Rules: Power rule, exponential rule, trigonometric integrals
- Substitution: Used when the integrand contains a function and its derivative
- Integration by Parts: Useful for products of functions
- Trigonometric Substitution: For integrals containing specific radical expressions
- Partial Fractions: For rational functions (polynomial fractions)
Applications of Integrals
Integrals have numerous applications in mathematics, physics, engineering, and other fields:
- Calculating areas, volumes, and lengths
- Determining work, energy, and power in physics
- Finding centers of mass and moments of inertia
- Solving differential equations
- Probability and statistics (probability density functions)