An integral calculator computes the antiderivative (indefinite integral) or definite integral of a mathematical function. For indefinite integrals, it finds F(x)+C where F'(x)=f(x). For definite integrals, it evaluates the net signed area under the curve f(x) between two bounds using the Fundamental Theorem of Calculus.
Integral Calculator
Our Integral Calculator handles both indefinite integrals (antiderivatives) and definite integrals (area under a curve). Enter an expression in terms of x and get the antiderivative instantly with all steps explained. For definite integrals, provide the lower and upper bounds to compute the exact numerical result. Supports polynomials, trigonometric functions (sin, cos, tan, sec, csc, cot), exponential (e^x), logarithms (ln x), square roots, and common composite functions. When symbolic integration is not possible, the calculator falls back to high-precision numerical integration using Simpson's rule.
functions Function
Use ^ for powers, * for multiply, sin/cos/exp/ln/sqrt for functions
Quick Examples
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functions Integration Rules
code Notation Guide
lightbulb Quick Tips
- •Use Definite to find area under the curve
- •Don't forget + C in indefinite integrals
- •∫(a·f) = a·∫f (constant factor rule)
- •∫f(ax+b) = F(ax+b)/a (linear substitution)
- •Verify: differentiate the antiderivative
How to Use This Calculator
Enter Function
Type your function using x as the variable. Use ^ for powers, * for multiplication, and standard function names like sin, cos, exp, ln, sqrt.
Choose Type
Select Indefinite for the general antiderivative (with +C), or Definite to compute the area under the curve between two bounds.
Set Bounds (Definite)
For definite integrals, enter the lower bound a and upper bound b. The result will be the net signed area F(b) - F(a).
View Steps
See which integration rules were applied (power rule, substitution, by parts) with each step explained clearly.
The Formula
Integration is the reverse of differentiation. The indefinite integral ∫f(x)dx gives a family of functions F(x)+C. The definite integral ∫[a,b]f(x)dx gives the net signed area under the curve from x=a to x=b, computed using the Fundamental Theorem of Calculus: F(b) - F(a).
∫f(x)dx = F(x) + C | ∫[a to b]f(x)dx = F(b) - F(a)
lightbulb Variables Explained
- f(x) The integrand — the function to integrate
- F(x) The antiderivative — a function whose derivative equals f(x)
- C Constant of integration for indefinite integrals
- a, b Lower and upper bounds for definite integrals
- dx The variable of integration (default: x)
tips_and_updates Pro Tips
Use ^ for powers: x^3 means x³. Use * for multiplication: 2*x or just 2x
Power rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (for n ≠ -1). Special case: ∫x⁻¹dx = ln|x| + C
Trig integrals: ∫sin(x)dx = -cos(x)+C, ∫cos(x)dx = sin(x)+C, ∫sec²(x)dx = tan(x)+C
Exponential: ∫eˣdx = eˣ+C, ∫aˣdx = aˣ/ln(a)+C
Logarithm: ∫ln(x)dx = x·ln(x) - x + C (use integration by parts)
For ∫f(ax+b)dx, substitute u=ax+b: result is F(ax+b)/a + C
This calculator supports: polynomials (x^2, 3*x^4), trigonometric functions (sin(x), cos(x), tan(x), sec(x), csc(x), cot(x)), exponentials (exp(x) or e^x), natural logarithm (ln(x)), square root (sqrt(x)), and composite functions. Use * for multiplication, ^ for exponentiation, and standard parentheses for grouping.
Frequently Asked Questions
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