∫ Integral Calculator

Calculate definite integrals and learn integration formulas with step-by-step explanations

Very Limited Function Support

This calculator currently supports ONLY:

  • Polynomials up to degree 3: Ax³ + Bx² + Cx + D
  • Definite integrals only (with upper and lower bounds)

NOT supported: Indefinite integrals, trigonometric functions (sin, cos, tan), exponential functions (eˣ), logarithms (ln x), rational functions, or any functions beyond cubic polynomials.

This is primarily an educational reference for integration formulas, not a full-featured integral calculator.

Definite Integral Calculator (Polynomial)

Calculate ∫[a to b] (Ax³ + Bx² + Cx + D) dx

Function:
x³ + 2x + 1
Definite Integral:
8.00

Common Integration Formulas

Power Functions

∫ xn dx = xn+1/(n+1) + C (n ≠ -1)
∫ 1/x dx = ln|x| + C
∫ 1 dx = x + C
∫ √x dx = (2/3)x3/2 + C

Exponential & Logarithmic

∫ ex dx = ex + C
∫ ax dx = ax/ln(a) + C
∫ ln(x) dx = x·ln(x) - x + C

Trigonometric

∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ sec²(x) dx = tan(x) + C
∫ csc²(x) dx = -cot(x) + C
∫ sec(x)tan(x) dx = sec(x) + C

Inverse Trigonometric

∫ 1/√(1-x²) dx = arcsin(x) + C
∫ 1/(1+x²) dx = arctan(x) + C
∫ 1/(x√(x²-1)) dx = arcsec(x) + C

How to Use This Calculator

  1. For the definite integral calculator, enter the coefficients of your polynomial function (A, B, C, D)
  2. Set your lower bound (a) and upper bound (b) for the limits of integration
  3. The calculator will display your function and compute the definite integral automatically
  4. Results update in real-time as you adjust any coefficient or bound
  5. For indefinite integrals, use the common formulas reference section below
  6. Remember to add the constant of integration C when finding antiderivatives
  7. Verify your answers by taking the derivative of your result

Understanding Integration

What is Integration?

Integration is the reverse process of differentiation. While derivatives tell us about rates of change and slopes, integrals tell us about accumulation and areas under curves. The integral symbol ∫ is an elongated S, representing "sum," because integration can be thought of as summing infinitely many infinitesimally small areas. Integration is fundamental to calculus and has applications in physics (calculating work, displacement), economics (consumer surplus), probability (finding probabilities from probability density functions), and countless other fields.

Definite vs. Indefinite Integrals

Indefinite Integral (∫f(x)dx): Represents a family of antiderivatives—functions whose derivative is f(x). Always includes "+ C" (constant of integration) since derivatives of constants are zero. Example: ∫x² dx = x³/3 + C. Definite Integral (∫[a to b]f(x)dx): Calculates the net signed area between the curve and the x-axis from x=a to x=b. Produces a number, not a function. Uses the Fundamental Theorem of Calculus: evaluate the antiderivative at b, then subtract its value at a.

The Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if F(x) is an antiderivative of f(x), then ∫[a to b]f(x)dx = F(b) - F(a). This powerful result means we can calculate definite integrals by finding antiderivatives rather than computing limits of sums. For example, to find ∫[0 to 3]x² dx, we find the antiderivative x³/3, evaluate it at 3 to get 27/3 = 9, evaluate at 0 to get 0, and subtract: 9 - 0 = 9.

Common Integration Techniques

Power Rule: The most basic technique—increase the exponent by 1 and divide by the new exponent. Substitution (u-substitution): The reverse of the chain rule; used when you have a function and its derivative present. Integration by Parts: Based on the product rule, using the formula ∫u dv = uv - ∫v du. Partial Fractions: Decomposes complex rational functions into simpler fractions. Trigonometric Substitution: Uses trig identities to simplify integrals containing square roots.

Tips for Successful Integration

1) Always check your answer by differentiating—you should get back the original function. 2) Memorize the common integration formulas, especially power rule, exponentials, and basic trig functions. 3) Look for patterns that suggest substitution: if you see f(g(x))·g'(x), try u = g(x). 4) For definite integrals, remember to evaluate at both bounds. 5) Practice recognizing which technique to use—this skill develops with experience. 6) Don't forget the constant C for indefinite integrals. 7) Draw graphs to visualize what the integral represents geometrically.

Frequently Asked Questions

What is the difference between definite and indefinite integrals?

An indefinite integral (∫f(x)dx) represents a family of antiderivatives and includes a constant C. It answers 'what function has this derivative?' A definite integral (∫[a to b]f(x)dx) calculates the net area under a curve between two bounds and produces a specific number, not a function.

What is the power rule for integration?

The power rule states: ∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1. For example, ∫x^3 dx = x^4/4 + C. This is the reverse of the power rule for differentiation and is the most fundamental integration technique.

How do you calculate a definite integral?

To calculate a definite integral ∫[a to b]f(x)dx: 1) Find the antiderivative F(x), 2) Evaluate F(b) - F(a). This is the Fundamental Theorem of Calculus. For example, ∫[0 to 2]x^2 dx = [x^3/3] from 0 to 2 = 8/3 - 0 = 8/3.

What are the most common integration formulas?

Common formulas include: ∫x^n dx = x^(n+1)/(n+1) + C, ∫e^x dx = e^x + C, ∫1/x dx = ln|x| + C, ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C, ∫sec^2(x) dx = tan(x) + C. These cover the majority of basic calculus integration problems.

Is this integral calculator free to use?

Yes! This integral calculator is completely free with no hidden costs, subscriptions, or limitations. Use it for homework, exam preparation, or learning calculus concepts.

Does this work on mobile devices?

Yes, this calculator is fully responsive and works perfectly on smartphones, tablets, and all desktop browsers.

Is my data private?

Absolutely. All processing happens locally in your browser. Your data never leaves your device and is not stored on any server.

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