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Integral Calculator

Evaluate indefinite integrals and definite integrals with a student-friendly workflow. This calculator shows the answer, explains the rule used, gives step-by-step support for common integral types, and includes a mini graph preview.

Background

An integral can represent accumulated change, area under a curve, or the reverse process of differentiation. For an indefinite integral, the goal is to find an antiderivative. For a definite integral, the goal is to compute the net signed area between a function and the x-axis over an interval.

Enter integral

Tip: Start with common forms like x^2, sin(x), or 1/x.

Supported syntax: x^2, x^2 + sin(x), cos(x), tan(x), e^x, e^(2x+1), sqrt(x), 1/(2x+3), x*e^x, x*sin(x), ln(x), pi, e.

Bounds

Bounds may use pi, e, decimals, and simple fractions like 1/2.

Chips prefill and calculate immediately.

Display options

Result

No results yet. Enter an integrand and click Calculate.

How to use this calculator

  • Choose Indefinite Integral to find an antiderivative and include + C.
  • Choose Definite Integral to evaluate the net signed area between two bounds.
  • Enter a function such as x^2, sin(x), e^x, or 1/x.
  • Use the quick-pick chips for common examples and to see supported formats instantly.

How this calculator works

  • Indefinite integrals: the calculator looks for a matching antiderivative rule for common forms.
  • Definite integrals: when a symbolic antiderivative is available, it applies the Fundamental Theorem of Calculus.
  • Numerical backup: if a symbolic form is not recognized, the calculator estimates the definite integral numerically.
  • Graph preview: the mini graph shows the curve and shades the interval for definite integrals.

Formula & Equations Used

Power rule: ∫ xn dx = xn+1/(n+1) + C, for n ≠ -1

Log rule: ∫ 1/x dx = ln|x| + C

Trig basics: ∫ sin(x) dx = -cos(x) + C, ∫ cos(x) dx = sin(x) + C

Exponential: ∫ ex dx = ex + C

Inverse trig form: ∫ 1/(1+x2) dx = arctan(x) + C

Definite integral: ∫ab f(x) dx = F(b) − F(a)

Example Problem & Step-by-Step Solution

Example 1 — Indefinite integral

Evaluate ∫ x² dx.

  1. Recognize a power of x.
  2. Use the power rule: add 1 to the exponent and divide by the new exponent.
  3. ∫ x² dx = x³/3 + C.

Example 2 — Definite integral

Evaluate ∫0π sin(x) dx.

  1. The antiderivative of sin(x) is −cos(x).
  2. Apply the bounds: [−cos(x)]0π.
  3. −cos(π) − (−cos(0)) = 1 − (−1) = 2.

Example 3 — u-sub style pattern

Evaluate ∫ (3x+1)^5 dx.

  1. Recognize the form (ax+b)^n.
  2. Use the pattern ∫ (ax+b)^n dx = (ax+b)^(n+1)/(a(n+1)) + C.
  3. ∫ (3x+1)^5 dx = (3x+1)^6/18 + C.

Frequently Asked Questions

Q: What is the difference between an indefinite and definite integral?

An indefinite integral gives a family of antiderivatives and includes + C. A definite integral gives a number over an interval.

Q: Does the calculator show exact answers?

For supported common symbolic forms, yes. For many definite integrals, it also shows a decimal approximation.

Q: Why can a definite integral be negative?

Because a definite integral measures signed area. Regions below the x-axis contribute negatively.

Q: Does this support every possible integral?

This version focuses on the most common student integrals and uses numerical estimation for many definite-integral cases.