Integral Calculator

Solve definite and indefinite integrals with detailed step-by-step working. Supports polynomial expressions, power rule integration, and area under curve calculations.

∫ Indefinite Integral ∫ₐᵇ Definite Integral (Area)

Function f(x)

Use ^ for exponents. Examples: 3x^2, 5x^3, 2x, 7, 4x^3 + 2x^2 + 3x + 5

Real-World Integral Examples

📈 Area Under a Parabola

Find the area under f(x) = x² from x = 0 to x = 3.

Indefinite integral: ∫ x² dx = x³/3 + C

Definite integral: ∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9 square units

The area bounded by the parabola y = x², the x-axis, and the lines x = 0 and x = 3 is 9 units².

📊 Velocity to Distance

A particle moves with velocity v(t) = 2t + 3 m/s. Find the distance traveled from t = 1 to t = 5.

Indefinite integral: ∫ (2t + 3) dt = t² + 3t + C

Definite integral: ∫₁⁵ (2t + 3) dt = [t² + 3t]₁⁵ = (25 + 15) − (1 + 3) = 36 meters

The integral of velocity gives displacement. The particle travels 36 meters between t = 1 and t = 5 seconds.

🔢 Power Rule: Cubic Function

Find the indefinite integral of f(x) = 4x³ + 3x² + 2x + 5

Integration: ∫ (4x³ + 3x² + 2x + 5) dx

Step 1: 4x³ → 4 · x⁴/4 = x⁴

Step 2: 3x² → 3 · x³/3 = x³

Step 3: 2x → 2 · x²/2 = x²

Step 4: 5 → 5x

Result: x⁴ + x³ + x² + 5x + C

📐 Area Between Curves

Find the area between f(x) = x² and the x-axis from x = −2 to x = 2.

∫ x² dx from −2 to 2

Antiderivative: x³/3

Evaluate: [x³/3]₋₂² = (8/3) − (−8/3) = 16/3 ≈ 5.333 square units

Since x² is symmetric and always positive, the area is twice the area from 0 to 2.

Understanding Integration

An integral is a mathematical operation that finds the area under a curve (definite integral) or the antiderivative of a function (indefinite integral). Integration is the inverse operation of differentiation.

Power Rule for Integration

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (where n ≠ −1)

For any real number n ≠ −1, increase the exponent by 1 and divide by the new exponent.

∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C

The special case when n = −1 gives the natural logarithm.

Basic Integration Rules

∫ k · f(x) dx = k · ∫ f(x) dx

Constant multiple rule: a constant factor can be pulled out of the integral.

∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

Sum/difference rule: the integral of a sum is the sum of the integrals.

Definite Integral (Fundamental Theorem of Calculus)

∫_a^b f(x) dx = F(b) − F(a)

Where F(x) is the antiderivative of f(x). The definite integral gives the signed area under the curve from x = a to x = b.

How to Integrate a Polynomial Step by Step

Identify each term: Break the polynomial into separate terms (e.g., 3x² + 2x + 1 → three terms)

Apply the power rule: For each term axⁿ, integrate to get a · xⁿ⁺¹/(n+1)

Simplify coefficients: Reduce fractions where possible (e.g., 3/3 = 1, 6/4 = 3/2)

Add the constant: For indefinite integrals, add + C (the constant of integration)

Evaluate bounds: For definite integrals, compute F(b) − F(a)

Quick Tips for Integration

📝 Check by Differentiating

You can verify your integration by differentiating the result. If you get back the original function, the integration is correct.

🔢 Watch the Exponent

The power rule fails when n = −1 (i.e., 1/x). In that case, the integral is ln|x| + C, not x⁰/0.

➕ Don't Forget +C

Indefinite integrals always include the constant of integration +C because the derivative of any constant is zero.

📐 Signed Area

Definite integrals give signed area. Area below the x-axis is negative. For total area, take the absolute value or split the integral at roots.

∫

Indefinite Integration

Find antiderivatives of polynomial functions using the power rule. Get the complete expression including the constant of integration +C.

∫ₐᵇ

Definite Integration

Calculate the exact signed area under a curve between any two bounds. Perfect for area problems, displacement calculations, and more.

📝

Step-by-Step Solutions

See the full integration process — each term integrated individually with the power rule, simplified, and combined into the final result.

🎓

Educational Tool

Designed for students learning calculus. Clear, annotated steps help you understand how integration works and verify your homework.

What Is an Integral?

An integral is a fundamental concept in calculus that represents the accumulation of quantities, such as areas under curves, total displacement from velocity, or the antiderivative of a function. Integration is the inverse operation of differentiation — together they form the core of calculus.

There are two main types of integrals: indefinite integrals (antiderivatives) which give a family of functions differing by a constant, and definite integrals which compute the signed area under a curve between two specific points. The definite integral is defined by the Fundamental Theorem of Calculus, which connects differentiation and integration.

For polynomial functions, integration follows the power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1). Each term is integrated separately, and the results are combined. The constant of integration +C is added for indefinite integrals, representing any constant value whose derivative would be zero.

Applications of Integration

Area under curves: Calculate the exact area bounded by a function and the x-axis between two points
Physics & Engineering: Displacement from velocity, work from force, charge from current
Probability: Finding probabilities from probability density functions (PDFs)
Economics: Consumer and producer surplus calculations using demand and supply curves
Volume of solids: Computing volumes of revolution using disk and shell methods

How to Integrate Polynomials

Integrating a polynomial is straightforward using the power rule and the linearity of integration. The integral of a sum is the sum of the integrals, and constant factors can be pulled out. Here's how it works:

Example: Integrate f(x) = 3x² + 2x + 1

Step 1: Separate into terms. The polynomial has three terms: 3x², 2x, and 1.

Step 2: Integrate each term using the power rule:

∫ 3x² dx = 3 · x³/3 = x³

∫ 2x dx = 2 · x²/2 = x²

∫ 1 dx = x (since 1 = x⁰ and x⁰⁺¹/(0+1) = x)

Step 3: Combine the results: x³ + x² + x + C

For a definite integral, evaluate the antiderivative at the upper bound and subtract the value at the lower bound. For example, ∫₀² (3x² + 2x + 1) dx = [x³ + x² + x]₀² = (8 + 4 + 2) − (0) = 14.

Common Integration Patterns

Constant term k: ∫ k dx = kx + C
Linear term ax: ∫ ax dx = ax²/2 + C
Quadratic term ax²: ∫ ax² dx = ax³/3 + C
Cubic term ax³: ∫ ax³ dx = ax⁴/4 + C
General term axⁿ: ∫ axⁿ dx = a · xⁿ⁺¹/(n+1) + C

Definite vs. Indefinite Integrals

The key difference between definite and indefinite integrals is that indefinite integrals produce a family of functions (antiderivatives) plus a constant of integration, while definite integrals produce a specific numerical value representing the signed area under the curve.

∫ Indefinite Integral

Written as ∫ f(x) dx = F(x) + C. Results in a function. Used to find antiderivatives and solve differential equations. Always includes +C.

∫ₐᵇ Definite Integral

Written as ∫ₐᵇ f(x) dx = F(b) − F(a). Results in a number. Used to compute areas, volumes, total change, and accumulated quantities.

🔗 Fundamental Theorem

The Fundamental Theorem of Calculus connects them: the definite integral of f from a to b equals F(b) − F(a), where F is any antiderivative of f.

🧮 Numerical Methods

For functions that are difficult to integrate analytically, numerical methods like the trapezoidal rule or Simpson's rule can approximate definite integrals.

Frequently Asked Questions

What is the difference between a definite and indefinite integral?

An indefinite integral (∫ f(x) dx) gives a family of antiderivative functions plus a constant of integration +C. It represents the general form of the antiderivative. A definite integral (∫ₐᵇ f(x) dx) gives a specific numerical value that represents the signed area under the curve f(x) from x = a to x = b. The definite integral is calculated by evaluating the antiderivative at the upper bound minus its value at the lower bound.

Why do indefinite integrals always have a +C?

The constant of integration +C is added because the derivative of any constant is zero. If F(x) is an antiderivative of f(x), then F(x) + 5, F(x) − 3, and F(x) + any constant all have the same derivative f(x). So the indefinite integral represents a family of functions differing only by a constant. When you differentiate the result, the constant disappears, so the +C accounts for all possible antiderivatives.

Can this calculator handle polynomials with negative exponents?

This calculator is designed for polynomial functions with non-negative integer exponents (e.g., 3x², 2x, 5, 4x³). It uses the power rule ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, which works for all n ≠ −1. Terms with negative exponents or terms like 1/x are not supported as polynomial inputs. For the special case of 1/x (n = −1), the integral would be ln|x| + C, which is outside the scope of this polynomial calculator.

What does the definite integral value represent?

The definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b. Areas above the x-axis contribute positive values, and areas below contribute negative values. If the function is entirely positive on [a, b], the integral gives the total area. If the function crosses the x-axis, the integral gives the net area (upper area minus lower area). For applications like displacement from velocity, the definite integral gives the net change in position.

How do I verify that my integration is correct?

The easiest way to verify an integration is to differentiate the result. If you differentiate F(x) + C and get back the original function f(x), your integration is correct. For example, if you integrate 3x² and get x³ + C, differentiate x³ + C to get 3x² — confirming the result. This relationship (differentiation and integration being inverse operations) is the essence of the Fundamental Theorem of Calculus.

What if my function has a fraction as a coefficient?

Fractional coefficients are handled exactly like any other constant. For example, ∫ (½)x² dx = ½ · x³/3 = x³/6 + C. Our calculator automatically simplifies fractional coefficients to their lowest terms. Just enter the function using decimal notation (e.g., 0.5x^2) or use the power rule as-is — the step-by-step solution will show all simplifications clearly.

⚠️ Important Note: This Integral Calculator is designed for educational and informational purposes. It handles polynomial integration using the power rule. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as engineering, physics, or financial decisions involving integration. Always consult a qualified professional for calculus-related decisions in high-stakes contexts.