Integral Calculator
Enter your function using standard notation: x^2, sin(x), cos(x), tan(x), e^x, ln(x), sqrt(x)
Quick Examples:
What is Integration?
Integration is the inverse of differentiation. It finds the antiderivative of a function or calculates the area under a curve between specified limits.
Integration is the inverse of differentiation. It finds the antiderivative of a function or calculates the area under a curve between specified limits.
Integration Rules & Tips
- Power Rule: ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
- Exponential: ∫e^x dx = e^x + C
- Trigonometric: ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C
- Logarithmic: ∫1/x dx = ln|x| + C
- Linearity: ∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx
Example: ∫(3x² + 2x + 1) dx = x³ + x² + x + C
Integration Result
Enter a function and click "Calculate Integral" to see the result here.
Step-by-Step Solution:
Related Math Calculators
How to Use the Integral Calculator
- Enter your function using standard mathematical notation (x^2, sin(x), e^x, ln(x), etc.)
- Choose between indefinite integral (antiderivative) or definite integral (area calculation)
- For definite integrals, specify the upper and lower limits
- Click "Calculate Integral" to see the result with step-by-step solution
- Use the quick example buttons to try common functions
Types of Integrals
- Indefinite Integrals: Find the general antiderivative F(x) + C
- Definite Integrals: Calculate the exact area under the curve between limits
- Improper Integrals: Integrals with infinite limits or discontinuous integrands
Integration Techniques
- Substitution Method: Used for composite functions
- Integration by Parts: For products of functions (∫u dv = uv - ∫v du)
- Partial Fractions: For rational functions
- Trigonometric Substitution: For expressions involving sqrt(a² ± x²)
Common Integration Applications
- Calculating areas under curves and between curves
- Finding volumes of solids of revolution
- Solving physics problems (velocity from acceleration, displacement from velocity)
- Computing work done by variable forces
- Calculating average values of functions
- Solving differential equations
Frequently Asked Questions about Integration
This calculator can solve both definite and indefinite integrals including polynomial, trigonometric, exponential, logarithmic, and rational functions. It handles most common functions encountered in calculus courses.
Enter functions using standard mathematical notation: x^2 for x squared, sin(x) for sine, cos(x) for cosine, tan(x) for tangent, e^x for exponential, ln(x) for natural logarithm, sqrt(x) for square root. Use parentheses to group terms clearly.
Indefinite integrals find the antiderivative (general form) and include a constant of integration (+C). Definite integrals calculate the exact area under a curve between specific limits and result in a numerical value.
Yes! The calculator provides detailed step-by-step solutions showing the integration technique used, intermediate steps, and the final result. This helps you understand the integration process and learn the methods.
The most important rules include: Power Rule (∫x^n dx = x^(n+1)/(n+1) + C), Exponential (∫e^x dx = e^x + C), Trigonometric (∫sin(x) dx = -cos(x) + C), Logarithmic (∫1/x dx = ln|x| + C), and the Linearity Property (∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx).
Integration has many practical applications: calculating areas and volumes, finding centers of mass, determining work done by variable forces, computing probabilities in statistics, analyzing electrical circuits, modeling population growth, and solving engineering problems involving rates of change.