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.
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.
- Recognize a power of x.
- Use the power rule: add 1 to the exponent and divide by the new exponent.
- ∫ x² dx = x³/3 + C.
Example 2 — Definite integral
Evaluate ∫0π sin(x) dx.
- The antiderivative of sin(x) is −cos(x).
- Apply the bounds: [−cos(x)]0π.
- −cos(π) − (−cos(0)) = 1 − (−1) = 2.
Example 3 — u-sub style pattern
Evaluate ∫ (3x+1)^5 dx.
- Recognize the form (ax+b)^n.
- Use the pattern ∫ (ax+b)^n dx = (ax+b)^(n+1)/(a(n+1)) + C.
- ∫ (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.
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