Integral Calculator: Compute Definite and Indefinite Integrals

Integration is the inverse operation of differentiation and a fundamental tool in calculus. It computes the area under a curve, the accumulated quantity from a rate of change, and the antiderivative of a function. Our Integral Calculator computes both definite integrals (with bounds) and indefinite integrals, showing step-by-step solutions using standard integration techniques.

What Is an Integral?

A definite integral ∫f(x)dx from a to b represents the signed area under the curve f(x) between x = a and x = b. An indefinite integral ∫f(x)dx represents the family of antiderivatives of f(x), written as F(x) + C where C is the constant of integration.

The Fundamental Theorem of Calculus connects differentiation and integration: if F(x) is the antiderivative of f(x), then ∫f(x)dx from a to b = F(b) - F(a). This means you can compute areas by finding antiderivatives.

Using the Integral Calculator

Enter your function using standard mathematical notation. For definite integrals, enter the lower and upper bounds. The calculator shows the integral, the step-by-step integration process, and the final result. Techniques supported include power rule, substitution (u-substitution), integration by parts, trigonometric integrals, and partial fractions.

The calculator also shows a shaded graph of the area being calculated for definite integrals, helping you visualize the region whose area you are computing.

Integration Techniques

Power Rule for Integration

∫x^n dx = x^(n+1)/(n+1) + C, for n ≠ -1. Example: ∫x³ dx = x^4/4 + C. For n = -1: ∫1/x dx = ln|x| + C.

U-Substitution

Used when the function contains a composition. Substitute u = g(x), du = g'(x)dx. Example: ∫2x × cos(x²)dx. Let u = x², du = 2x dx. The integral becomes ∫cos(u)du = sin(u) + C = sin(x²) + C.

Integration by Parts

Used for products of functions: ∫u dv = uv - ∫v du. Example: ∫x × e^x dx. Let u = x, dv = e^x dx. Then du = dx, v = e^x. Result: x × e^x - ∫e^x dx = x × e^x - e^x + C = e^x(x - 1) + C.

Applications of Integration

• Area calculation: Find the area between curves, under a curve, or bounded by functions. Used in geometry, physics, and engineering.
• Volume of revolution: Rotate a function around an axis to find the volume of the resulting solid. Disk and shell methods are both supported.
• Physics: Integrate velocity to get position, acceleration to get velocity, force over distance to get work, and charge flow to get total charge.
• Probability: Integrate probability density functions to find cumulative probabilities and expected values.
• Economics: Integrate marginal cost to get total cost, marginal revenue to get total revenue, and calculate consumer and producer surplus.

Real-World Example

Find the area under f(x) = x² from x = 1 to x = 3:

• Indefinite integral: ∫x² dx = x³/3 + C
• Definite integral: ∫₁³ x² dx = [x³/3]₁³ = (27/3) - (1/3) = 9 - 0.333 = 8.667
• Area: 8.667 square units

This could represent the total distance traveled if x² represents velocity over time, the total cost if x² represents marginal cost, or the accumulated growth if x² represents a growth rate.

Start Calculating

Use our Integral Calculator below to compute integrals and understand accumulated change. Also check our Derivative Calculator for rates of change and our Area Calculator for geometric area calculations.