Integral Calculator 2025 - Free Steps & Antiderivatives

Solve integrals instantly with our free antiderivative calculator. Get step-by-step solutions for definite and indefinite integrals. Mastering calculus has never been easier!

✅ Indefinite & Definite ⚡ Instant Solutions 📚 Step-by-Step 📱 Mobile Friendly

🎯 What is an Integral?

An integral is a fundamental concept in calculus used to find areas, volumes, and central points. Integration is essentially the reverse process of differentiation. While a derivative finds the rate of change, an integral finds the quantity that accumulated to produce that rate. Our Integral Calculator helps you solve both indefinite integrals (antiderivatives) and definite integrals (area calculations).

Fundamental Theorem of Calculus:

∫[a,b] f(x) dx = F(b) - F(a)

Where F(x) is the antiderivative of f(x).

🧮 Solve Your Integral Problem

Variable:

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📊 Integration Result

Input Function
Antiderivative

📝 Solution Steps

📖 What is an Integral? (The Complete Guide)

An integral is one of the two core concepts in calculus (the other being the derivative). While differentiation calculates the rate of change, **integration** calculates the accumulation of quantities. Mathematically, it is the reverse process of differentiation, often referred to as finding the antiderivative.

In geometric terms, a definite integral represents the precise numerical area under a curve bounded by the x-axis and vertical lines. This capability is crucial for solving real-world problems in physics, engineering, and economics, such as calculating distance traveled from velocity or finding the center of mass.

🧩 Definite vs. Indefinite Integrals

Indefinite Integral (Antiderivative)

Written as ∫ f(x) dx.

It results in a **function** (a family of curves). You must always add the constant of integration (+ C) because the derivative of a constant is zero.

Example: ∫ 2x dx = x² + C

Definite Integral (Area)

Written as ∫[a,b] f(x) dx.

It results in a specific **number**. It calculates the net area between x = a and x = b using the Fundamental Theorem of Calculus: F(b) - F(a).

Example: ∫[0,2] 2x dx = 4

📜 A Brief History of Integration

The origins of integration can be traced back to ancient Greece, where mathematicians like Archimedes used the method of exhaustion to calculate the areas and volumes of geometric shapes. However, it wasn't until the 17th century that Isaac Newton and Gottfried Wilhelm Leibniz independently developed the Fundamental Theorem of Calculus, which established the inverse relationship between differentiation and integration.

Today, the symbol ∫ (an elongated 'S' for summa) remains a testament to Leibniz's contribution to the field. For a deeper dive into the relationship between these two branches, check out our guide on Integration vs. Differentiation.

🎯 How to Use the Integral Calculator

Our tool is designed to be the most user-friendly symbolic integration solver on the web. Follow these simple steps:

  1. Enter Function: Type your expression in the input box (e.g., x^2 + sin(x)). Use standard math notation.
  2. Select Variable: Choose the variable of integration (usually x, but we support y, t, and θ).
  3. Toggle Definite/Indefinite:
    • Keep the toggle OFF for indefinite integrals (finding the antiderivative).
    • Turn the toggle ON to calculate the area. Enter your "Lower Limit" and "Upper Limit".
  4. Calculate: Press the "Calculate Integral" button.
  5. Review Steps: Analyze the step-by-step breakdown to learn properly.

📜 Comprehensive Integration Rules Table

Memorizing these standard forms will speed up your calculus problem solving. Our calculator uses these exact rules to generate steps.

Rule Name Formula Example
Power Rule ∫ x^n dx = x^(n+1)/(n+1) + C ∫ x² dx = x³/3 + C
Constant Multiple ∫ a·f(x) dx = a · ∫ f(x) dx ∫ 5x dx = 5 · (x²/2)
Sum/Difference ∫ (u ± v) dx = ∫ u dx ± ∫ v dx ∫ (x+1) dx = x²/2 + x
Exponential ∫ e^x dx = e^x + C -
Logarithmic ∫ 1/x dx = ln|x| + C -
Trigonometric ∫ cos(x) dx = sin(x) + C -

🌐 Real-World Applications of Integration

Why do we learn integrals? It's not just for math method exams. Integration is the language of accumulation and is used everywhere:

🚀 Physics & Engineering

Calculating displacement from velocity, work done by a variable force, center of mass, and electricity concepts like charge accumulation.

📈 Economics

Determining Consumer Surplus and Producer Surplus by integrating supply and demand curves. Finding total revenue from marginal revenue.

🩺 Biology & Medicine

Modeling population growth over time, blood flow rates through arteries, and cardiac output calculations.

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We believe education should be free. See how we stack up against the paid giants:

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❓ Frequently Asked Questions (FAQ)

How do I integrate e^x^2?

The integral of e^(x^2) does not have an elementary antiderivative. It involves the imaginary error function (erfi). Most standard calculators will return a non-elementary result or an approximation.

What is the "Area under Curve"?

The area under a curve between two points x=a and x=b is given by the definite integral from a to b. If the curve drops below the x-axis, that area counts as negative.

Can I use U-Substitution?

Yes! Our calculator automatically detects when Integration by Substitution (u-sub) is needed and will show the substitution steps in the solution.