What is the difference between definite and indefinite integrals?

An indefinite integral ∫f(x)dx gives the antiderivative F(x)+C — a family of functions. An definite integral ∫[a,b]f(x)dx gives a number representing the net signed area under the curve from x=a to x=b. By the Fundamental Theorem of Calculus, the definite integral equals F(b)-F(a) where F is any antiderivative.

What is the power rule for integration?

The power rule states: ∫xⁿdx = xⁿ⁺¹/(n+1) + C, for any n ≠ -1. For example, ∫x³dx = x⁴/4 + C. The special case n = -1 gives ∫x⁻¹dx = ∫(1/x)dx = ln|x| + C.

How do you integrate trigonometric functions?

Basic trig integrals: ∫sin(x)dx = -cos(x)+C, ∫cos(x)dx = sin(x)+C, ∫tan(x)dx = -ln|cos(x)|+C, ∫sec²(x)dx = tan(x)+C, ∫csc²(x)dx = -cot(x)+C. For ∫sin(ax)dx = -cos(ax)/a+C (linear substitution).

What is integration by substitution?

Substitution (u-substitution) replaces a complex expression with u to simplify. If the integrand has the form f(g(x))·g'(x), let u=g(x), du=g'(x)dx. The integral becomes ∫f(u)du. Example: ∫2x·cos(x²)dx — let u=x², du=2x dx → ∫cos(u)du = sin(u)+C = sin(x²)+C.

What is integration by parts?

Integration by parts: ∫u·dv = u·v - ∫v·du. Choose u and dv using LIATE order (Logarithm, Inverse trig, Algebraic, Trig, Exponential) — pick u from earlier in the list. Example: ∫x·eˣdx — u=x, dv=eˣdx → du=dx, v=eˣ → x·eˣ - ∫eˣdx = x·eˣ - eˣ + C = eˣ(x-1)+C.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus (FTC) has two parts: (1) If F(x)=∫[a,x]f(t)dt, then F'(x)=f(x) — differentiation and integration are inverse operations. (2) ∫[a,b]f(x)dx = F(b)-F(a) where F is any antiderivative of f. This connects the geometric concept of area with antiderivatives.

How does numerical integration work?

When symbolic integration is difficult, numerical methods approximate the definite integral. Simpson's Rule divides [a,b] into n equal intervals and approximates each piece with a parabola: ∫[a,b]f(x)dx ≈ (h/3)[f(x₀)+4f(x₁)+2f(x₂)+4f(x₃)+...+f(xₙ)], where h=(b-a)/n. This is more accurate than the Trapezoidal Rule (O(h⁴) error vs O(h²)).

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.

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calendar_today Updated: April 2026

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functions Integration Rules

Power Rule

∫xⁿdx = xⁿ⁺¹/(n+1) + C

1/x Rule

∫(1/x)dx = ln|x| + C

Exponential

∫eˣdx = eˣ + C

Sine

∫sin(x)dx = −cos(x) + C

Cosine

∫cos(x)dx = sin(x) + C

Natural Log

∫ln(x)dx = x·ln(x)−x + C

FTC (Definite)

∫[a,b] = F(b) − F(a)

code Notation Guide

Power x^2, x^(1/3)

Multiply 3*x or 3x

Trig sin(x), cos(x)

Exp / Log exp(x), ln(x)

Square Root sqrt(x)

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

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.

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.

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