∫ Integral Calculator

Calculate definite integrals numerically using Simpson's Rule, Trapezoidal Rule, and Midpoint Rule with step-by-step solutions.

Definite Integral — Simpson's Rule

Function f(x)

Number of Intervals (even)

Lower Limit a

Upper Limit b

∫ f(x) dx ≈ (Simpson's Rule)

Definite Integral — Trapezoidal Rule

Function f(x)

Number of Intervals n

Lower Limit a

Upper Limit b

∫ f(x) dx ≈ (Trapezoidal Rule)

Definite Integral — Midpoint Rule

Function f(x)

Number of Rectangles n

Lower Limit a

Upper Limit b

∫ f(x) dx ≈ (Midpoint Rule)

Common Integral Formulas Reference

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ −1)

∫ 1/x dx = ln|x| + C

∫ eˣ dx = eˣ + C

∫ sin(x) dx = −cos(x) + C

∫ cos(x) dx = sin(x) + C

∫ ln(x) dx = x·ln(x) − x + C

∫ √x dx = (2/3)x^(3/2) + C

Accuracy Notes

Simpson's Rule is generally most accurate (4th-order error). Trapezoidal is 2nd-order. More intervals (n) = higher accuracy for all methods.

Simpson's error ≈ −(b−a)⁵/(180n⁴) × f⁴(ξ)
Trapezoidal error ≈ −(b−a)³/(12n²) × f″(ξ)

How the Integral Calculator Works

How Integration Works

Integration finds the area under a curve by dividing it into small sections and summing their areas. This calculator uses numerical methods to approximate definite integrals when analytical solutions are difficult.

Mathematical Formulas:

Simpson’s Rule:
∫[a,b] f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ)]
Most accurate for smooth functions

Trapezoidal Rule:
∫[a,b] f(x)dx ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + … + f(xₙ)]
Good balance of accuracy and simplicity

Rectangular Rule:
∫[a,b] f(x)dx ≈ h[f(x₁) + f(x₂) + … + f(xₙ)]
Basic approximation method

Real-World Applications:

Physics: Calculate work done by variable forces, center of mass
Engineering: Find areas of irregular shapes, fluid flow analysis
Economics: Consumer surplus, producer surplus calculations
Statistics: Probability density function areas, normal distribution
Medicine: Drug concentration over time, bioavailability studies

Usage Recommendations:

Use Simpson’s Rule for smooth, continuous functions (most accurate)
Choose Trapezoidal Rule for general-purpose calculations
Increase intervals (n) for higher accuracy, especially for oscillating functions
For functions like x², x³, sin(x), cos(x), use at least 10-20 intervals
Verify results by comparing different methods when precision is critical