Integral Calculator
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Step-by-Step Solution
Result
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Step-by-Step Solution
Common Integration Formulas
Calculation History
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Powerful Integration Features
Our calculator provides advanced features to make integration simple and educational
Instant Calculation
Get immediate results for both simple and complex integrals with our powerful engine
Step-by-Step Solutions
Learn the integration process with detailed explanation for each step
Indefinite & Definite
Calculate both indefinite integrals and definite integrals with bounds
Mobile Friendly
Use our calculator on any device with a responsive interface that adapts to your screen
Calculation History
Keep track of your recent calculations for easy reference and comparison
Dark/Light Mode
Choose between light and dark themes for comfortable viewing day or night
How to Use Our Calculator
Follow these simple steps to calculate integrals
Choose Integral Type
Select between indefinite or definite integration depending on your needs
Enter Expression
Type your mathematical expression using the keyboard or the buttons provided
Add Limits (if needed)
For definite integrals, enter the upper and lower bounds of integration
Calculate
Click the Calculate button to compute the integral and view the step-by-step solution
Pro Tips
- Use ^ for exponents (e.g., x^2 for x²)
- Use * for multiplication (e.g., 2*x for 2x)
- Use parentheses for grouping expressions
- For trigonometric functions, use sin(x), cos(x), etc.
- For complex functions, try our new Complex tab for advanced integration techniques
Multiple Solution Approaches
Explore different methods to solve the same integral with varying complexity levels
∫ x·sin(x) dx
Integration by Parts
1. Let u = x and dv = sin(x)dx
2. Then du = dx and v = -cos(x)
3. Apply the formula: ∫u·dv = u·v - ∫v·du
4. ∫x·sin(x)dx = x·(-cos(x)) - ∫(-cos(x))dx
5. = -x·cos(x) + ∫cos(x)dx
6. = -x·cos(x) + sin(x) + C
7. = sin(x) - x·cos(x) + C
∫ e^x·sin(x) dx
Double Integration by Parts
1. Let u = e^x and dv = sin(x)dx
2. Then du = e^x·dx and v = -cos(x)
3. Apply the formula: ∫u·dv = u·v - ∫v·du
4. ∫e^x·sin(x)dx = e^x·(-cos(x)) - ∫(-cos(x))·e^x·dx
5. = -e^x·cos(x) + ∫e^x·cos(x)dx
6. For ∫e^x·cos(x)dx, use integration by parts again
7. Let u = e^x and dv = cos(x)dx
8. Then du = e^x·dx and v = sin(x)
9. ∫e^x·cos(x)dx = e^x·sin(x) - ∫e^x·sin(x)dx
10. Substituting back: ∫e^x·sin(x)dx = -e^x·cos(x) + e^x·sin(x) - ∫e^x·sin(x)dx
11. 2∫e^x·sin(x)dx = -e^x·cos(x) + e^x·sin(x) + C
12. ∫e^x·sin(x)dx = (e^x·(sin(x)-cos(x)))/2 + C
∫ 1/(1+x^2) dx
Standard Form Recognition
1. Recognize that 1/(1+x^2) is the derivative of arctan(x)
2. Apply the formula: ∫1/(1+x^2)dx = arctan(x) + C
∫ x·ln(x) dx
Integration by Parts
1. Let u = ln(x) and dv = x·dx
2. Then du = (1/x)·dx and v = x^2/2
3. Apply the formula: ∫u·dv = u·v - ∫v·du
4. ∫x·ln(x)dx = ln(x)·(x^2/2) - ∫(x^2/2)·(1/x)dx
5. = (x^2·ln(x))/2 - ∫(x/2)dx
6. = (x^2·ln(x))/2 - x^2/4 + C
∫ sec^3(x) dx
Reduction Formula
1. Use the reduction formula: ∫sec^n(x)dx = (sec^(n-2)(x)·tan(x))/(n-1) + ((n-2)/(n-1))·∫sec^(n-2)(x)dx
2. For n = 3: ∫sec^3(x)dx = (sec(x)·tan(x))/2 + (1/2)·∫sec(x)dx
3. We know that ∫sec(x)dx = ln|sec(x) + tan(x)| + C
4. Therefore: ∫sec^3(x)dx = (sec(x)·tan(x))/2 + (1/2)·ln|sec(x) + tan(x)| + C
∫ √(1-x^2) dx
Trigonometric Substitution
1. Let x = sin(θ), then dx = cos(θ)dθ
2. √(1-x^2) = √(1-sin^2(θ)) = √(cos^2(θ)) = |cos(θ)| = cos(θ) for θ in [-π/2, π/2]
3. ∫√(1-x^2)dx = ∫cos(θ)·cos(θ)dθ = ∫cos^2(θ)dθ
4. Using the identity cos^2(θ) = (1 + cos(2θ))/2
5. ∫cos^2(θ)dθ = ∫(1 + cos(2θ))/2 dθ = θ/2 + sin(2θ)/4 + C
6. Since x = sin(θ), we have θ = arcsin(x)
7. Also, sin(2θ) = 2sin(θ)cos(θ) = 2x·√(1-x^2)
8. Therefore: ∫√(1-x^2)dx = (arcsin(x) + x·√(1-x^2))/2 + C
Frequently Asked Questions
Find answers to common questions about our integral calculator
How accurate is this calculator?
Our calculator uses advanced mathematical algorithms to provide highly accurate results for a wide range of integrals. However, for extremely complex expressions or special cases, we recommend verifying the results with alternative methods.
Can I use this calculator for my homework?
Our calculator is designed to be educational by providing step-by-step solutions. This helps you understand the integration process while checking your work. We encourage using it as a learning tool rather than just for getting answers.
What types of integrals can this calculator solve?
Our calculator can handle a wide variety of integrals including polynomials, trigonometric functions, exponentials, logarithms, and combinations of these. Both indefinite integrals and definite integrals with numerical bounds are supported.
Is there a limit to the complexity of expressions?
While our calculator can handle most standard integrals, extremely complex expressions or those requiring specialized techniques might be challenging. If you encounter an expression our calculator cannot solve, try breaking it down into simpler parts.
Do I need to create an account to use the calculator?
No, our integral calculator is completely free to use without any registration required. However, creating an account allows you to save your calculation history and access it from any device.
Can I use this calculator on my mobile device?
Yes, our calculator is fully responsive and works on all devices including smartphones, tablets, and desktop computers. The interface adapts to provide the best experience on any screen size.
Essential Integration Formulas
Master these key formulas to solve integration problems efficiently
Basic Integrals
Trigonometric Integrals
Exponential Integrals
Logarithmic Integrals
Integration by Parts
Where u(x) and v(x) are functions of x, and u'(x) and v'(x) are their derivatives. This technique is useful for integrals like ∫ x·sin(x) dx or ∫ x·ex dx.
