🧮 Math Calculator

Integral Calculator

Calculate definite and indefinite integrals of common functions with step-by-step solutions. Perfect for calculus students, teachers, and professionals working with mathematical analysis.

Integral Calculator

Calculate definite and indefinite integrals of common functions with step-by-step solutions

Select Function Type

Function Type:
Quadratic function: ax² + bx + c
Indefinite Integral

Common Integration Rules

• Power Rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

• Constant Rule: ∫cdx = cx + C

• Exponential: ∫eˣdx = eˣ + C

• Logarithmic: ∫(1/x)dx = ln|x| + C

• Trigonometric: ∫sin(x)dx = -cos(x) + C

Common Integrals

∫x²dx
= x³/3 + C
∫x³dx
= x⁴/4 + C
∫eˣdx
= eˣ + C
∫ln(x)dx
= xln(x) - x + C
∫sin(x)dx
= -cos(x) + C
∫cos(x)dx
= sin(x) + C

Understanding Integrals

Integration is the reverse process of differentiation. While derivatives give us the rate of change of a function, integrals give us the accumulated change or the area under a curve.

Indefinite integrals (∫f(x)dx) represent the family of all antiderivatives of a function, including an arbitrary constant C. These are useful for finding general solutions to differential equations.

Definite integrals (∫ₐᵇf(x)dx) give us the exact value of the area under the curve between two points a and b. They have numerous applications in physics, engineering, and economics.

How to Use This Calculator

1Choose between indefinite and definite integration
2Select the type of function you want to integrate
3Enter the required coefficients and parameters
4For definite integrals, enter the bounds a and b
5Click "Calculate Integral" to see the result and steps

Key Integration Rules

⚡

Power Rule

For f(x) = xⁿ (n ≠ -1), the integral is ∫xⁿdx = xⁿ⁺¹/(n+1) + C. This is the reverse of the power rule for derivatives.

📐

Trigonometric

∫sin(x)dx = -cos(x) + C, ∫cos(x)dx = sin(x) + C, ∫tan(x)dx = -ln|cos(x)| + C.

📊

Exponential & Log

∫eˣdx = eˣ + C, ∫(1/x)dx = ln|x| + C, ∫aˣdx = aˣ/ln(a) + C.