Integral Calculator – Solve Definite and Indefinite Integrals Online
Calculate integrals step-by-step with our free online integral calculator. Solve both definite and indefinite integrals instantly with detailed solutions.
Online Integral Calculator
What is an Integral Calculator?
An integral calculator is a mathematical tool that computes the integral of a given function. Integration is the reverse process of differentiation and is used to find areas under curves, volumes, and solve various mathematical problems in calculus.
Types of Integrals
- Indefinite Integrals: Also called antiderivatives, these represent a family of functions whose derivative is the given function
- Definite Integrals: These calculate the exact numerical value of the area under a curve between two specific points
How to Use the Integral Calculator
- Enter your mathematical function in the input field
- Specify the variable of integration (usually x)
- Choose between indefinite or definite integral
- For definite integrals, enter the lower and upper limits
- Click “Calculate Integral” to get your result
Supported Mathematical Functions
| Function | Syntax | Example |
|---|---|---|
| Power | x^n | x^2, x^3 |
| Sine | sin(x) | sin(x), sin(2*x) |
| Cosine | cos(x) | cos(x), cos(x^2) |
| Natural Log | ln(x) | ln(x), ln(x+1) |
| Exponential | exp(x) | exp(x), exp(2*x) |
Integration Rules and Formulas
Basic Integration Rules
- Power Rule: ∫ x^n dx = x^(n+1)/(n+1) + C (where n ≠ -1)
- Constant Rule: ∫ k dx = kx + C
- Sum Rule: ∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx
- Constant Multiple Rule: ∫ k·f(x) dx = k·∫ f(x) dx
Common Integral Formulas
- ∫ 1/x dx = ln|x| + C
- ∫ e^x dx = e^x + C
- ∫ sin(x) dx = -cos(x) + C
- ∫ cos(x) dx = sin(x) + C
- ∫ sec²(x) dx = tan(x) + C
Applications of Integration
Integration has numerous practical applications across various fields:
Mathematics and Physics
- Calculating areas under curves
- Finding volumes of solids of revolution
- Determining work done by variable forces
- Computing centers of mass and moments of inertia
Engineering and Technology
- Signal processing and Fourier analysis
- Calculating flow rates and accumulated quantities
- Structural analysis and load calculations
- Probability and statistics applications
Tips for Using the Integral Calculator
- Always use parentheses to clarify the order of operations
- Use * for multiplication (write 2*x instead of 2x)
- For complex functions, break them down into simpler parts
- Verify your results by differentiating the answer
- Double-check the limits for definite integrals
Frequently Asked Questions
What’s the difference between definite and indefinite integrals?
Indefinite integrals represent a family of functions (antiderivatives) and include a constant of integration (+C). Definite integrals calculate a specific numerical value representing the area under a curve between two points.
Can this calculator solve all types of integrals?
Our calculator can solve most common polynomial, trigonometric, exponential, and logarithmic functions. For very complex functions, consider breaking them into simpler parts or consulting advanced mathematical software.
How accurate are the calculations?
The calculator provides highly accurate results for supported functions. For numerical approximations of definite integrals, the accuracy depends on the complexity of the function and the integration method used.