Integral Calculator
Evaluate definite integrals (∫ₐᵇ f(x) dx) and indefinite antiderivatives (F(x) + C) step-by-step with live 2D Riemann area under the curve shading.
Integral Settings
Step-by-Step Fundamental Theorem of Calculus Evaluation
How to Calculate an Integral
To evaluate a definite integral ∫ₐᵇ f(x) dx: 1. Find an antiderivative function F(x) such that F'(x) = f(x). 2. Apply the Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a). For indefinite integrals, express the family of all antiderivatives as F(x) + C.
The Fundamental Theorem of Calculus (FTC)
The Fundamental Theorem of Calculus unites the two primary branches of analysis—differential calculus (rates of change) and integral calculus (accumulated area)—demonstrating that they are exact mathematical inverses:
Where F(x) is any antiderivative satisfying F'(x) = f(x). This eliminates the need to compute infinite Riemann limits manually for elementary continuous functions.
Definite vs. Indefinite Integrals & The Constant of Integration (C)
Returns a family of functions. The constant +C arises because the derivative of any constant is zero (d/dx[C] = 0).
Returns a single numerical scalar value. Regions above the x-axis contribute positive area; regions below contribute negative area.
Core Integration Techniques
| Method | Formula / Rule | Primary Use Case |
|---|---|---|
| Power Rule | ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C | Polynomials & radicals (n ≠ −1) |
| Logarithmic | ∫ (1/x) dx = ln|x| + C | Reciprocal terms (n = −1) |
| U-Substitution | ∫ f(g(x))g'(x) dx = ∫ f(u) du | Reverse of the Chain Rule |
| By Parts | ∫ u dv = uv − ∫ v du | Products of polynomial & trig/exp functions |
Riemann Sums & Geometric Area Under the Curve
Bernhard Riemann formalized integration in 1854 as the infinite limit of step approximations:
As the number of rectangular subintervals n grows toward infinity, the approximation error shrinks to zero, yielding the continuous exact integral.
Real-World Applications of Integral Calculus
Mechanical Work & Physics
Calculating mechanical work done by variable forces $W = \int F(x) dx$, rocket launch escape energy, and centers of mass.
Probability & Statistics
Integrating continuous probability density functions (PDFs) to compute cumulative distribution probabilities $P(a \le X \le b) = \int_a^b f(x) dx$.
Civil & Fluid Engineering
Hydrostatic pressure forces on dams and total water storage reservoir volumes calculated via double and triple integrals.
Step-by-Step Worked Numerical Solutions
Problem: Evaluate ∫₀² (3x² + 2x − 1) dx.
Common Pitfalls & Mistakes
Forgetting the Constant of Integration (+C)
Omitting +C on indefinite integrals represents a loss of all potential family solutions.
Sign Reversal on Trig Antiderivatives
∫ sin(x) dx = −cos(x) + C (with a negative sign), whereas d/dx[sin(x)] = +cos(x).
Applying Power Rule to 1/x
Setting n = −1 in xⁿ⁺¹/(n+1) yields division by zero (x⁰/0). The correct integral is ln|x| + C.
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