Free integral calculator with step-by-step solutions. Compute indefinite and definite integrals with interactive graph, common formulas, and numerical methods.
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Integral Calculator, Math, Free integral calculator with step-by-step solutions. Compute indefinite and definite integrals with interactive graph, common formulas, and numerical methods., antiderivative, integration, calculus solver, definite integral, calc, compute
Integral Calculator
Free integral calculator with step-by-step solutions. Compute indefinite and definite integrals with interactive graph, common formulas, and numerical methods.
antiderivative, integration, calculus solver, definite integral
Math global
Integral Calculator, Math, Free integral calculator with step-by-step solutions. Compute indefinite and definite integrals with interactive graph, common formulas, and numerical methods., antiderivative, integration, calculus solver, definite integral, calc, compute
Integral Calculator
Free integral calculator with step-by-step solutions. Compute indefinite and definite integrals with interactive graph, common formulas, and numerical methods.
Integration
∫ x^2 + 3x + 1 dx
Antiderivative
x^3/3 + 3 · (x^2/2) + 1x + C
Sum/Difference Rule
Function Graph
Graph of f(x) = x^2 + 3x + 1
f(x)
Step-by-Step Solution
See how the integral is computed
1Split the integral
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
2Left: Split the integral
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
3Left: Left: Power Rule
∫ x^n dx = x^(n+1)/(n+1) + C
n = 2, n+1 = 3
x^3/3 + C
4Left: Right: Factor out constant
∫ 3 · f(x) dx = 3 · ∫ f(x) dx
5Left: Right: Power Rule
∫ x dx = x^(1+1)/(1+1) + C
n = 1
x^2/2 + C
6Left: Right: Multiply by constant
3 · (x^2/2) + C
7Left: Combine results
x^3/3 + 3 · (x^2/2) + C
8Right: Constant Rule
∫ c dx = cx + C
c = 1
1x + C
9Combine results
x^3/3 + 3 · (x^2/2) + 1x + C
Common Integral Formulas
Quick reference table of standard integrals
∫ xⁿ dx
= xⁿ⁺¹/(n+1) + C
(n ≠ -1)
∫ 1/x dx
= ln|x| + C
∫ eˣ dx
= eˣ + C
∫ aˣ dx
= aˣ/ln(a) + C
(a > 0)
∫ 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
∫ ln(x) dx
= x·ln(x) - x + C
What Is an Integral?
The fundamental concept of calculus that reverses differentiation
An integral computes the accumulation of quantities. In calculus, integration is the reverse process of differentiation. If the derivative tells you the rate of change, the integral tells you the total accumulated change.
There are two types: the indefinite integral (antiderivative) finds a family of functions whose derivative is the given function, while the definite integral computes the net signed area between a function and the x-axis over an interval.
Fundamental Theorem of Calculus
If F'(x) = f(x), then ∫f(x) dx = F(x) + C
∫ab f(x) dx = F(b) - F(a)
Common Integration Techniques
Methods for finding antiderivatives of different function types
Power Rule
∫ xn dx = xn+1/(n+1) + C (n ≠ -1)
The most fundamental integration rule. Increase the exponent by 1 and divide by the new exponent.
U-Substitution
When the integrand contains a composite function f(g(x)) and its derivative g'(x) appears as a factor, substitute u = g(x) to simplify. This is the reverse of the chain rule.
Example
∫ 2x cos(x²) dx
Let u = x², du = 2x dx
= ∫ cos(u) du = sin(u) + C = sin(x²) + C
Integration by Parts
∫ u dv = uv - ∫ v du
Used when the integrand is a product of two functions. Choose u (to differentiate) and dv (to integrate) using the LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
Definite vs. Indefinite Integrals
Understanding the two types of integrals and when to use each
Indefinite Integral
Finds a family of antiderivatives
Result includes + C (constant)
No bounds of integration
Definite Integral
Computes a numerical value
Result is F(b) - F(a)
Represents net signed area
Common Integration Mistakes to Avoid
Frequent errors and how to prevent them
Forgetting + C
Indefinite integrals always include a constant of integration. Omitting it loses a family of valid solutions.
Wrong power rule for n = -1
The power rule doesn't work for ∫ x⁻¹ dx. This integral equals ln|x| + C, not x⁰/0.
Missing chain rule factor
When integrating sin(3x), the result is -cos(3x)/3, not -cos(3x). Don't forget to divide by the inner derivative.
Treating ∫ as distributive over products
∫ f(x)g(x) dx ≠ (∫ f dx)(∫ g dx). Integration is linear only for sums, not products.
Frequently Asked Questions
Common questions about integrals and integration techniques
An integral is the reverse of differentiation. The indefinite integral (antiderivative) finds a function whose derivative equals the given function. The definite integral computes the net signed area between a function and the x-axis over a specified interval [a, b]. Together, they form one of the two fundamental operations of calculus.
An indefinite integral has no bounds and produces a function plus a constant of integration (+C), representing a family of antiderivatives. A definite integral has upper and lower bounds and produces a single numerical value, computed as F(b) - F(a) where F is any antiderivative of the integrand.
The constant of integration +C appears because differentiation eliminates constants (the derivative of any constant is zero). So if F'(x) = f(x), then (F(x) + C)' = f(x) for any constant C. The +C represents all possible antiderivatives that differ by a constant.
Start with basic rules: power rule for polynomials, standard formulas for trig/exponential functions. If the integrand is a composition, try u-substitution. For products of functions, try integration by parts (LIATE rule). For rational functions, try partial fractions. For expressions with square roots of quadratics, try trigonometric substitution.
The power rule states that ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, where n ≠ -1. You increase the exponent by 1 and divide by the new exponent. The special case n = -1 gives ∫ 1/x dx = ln|x| + C.
U-substitution is the integration equivalent of the chain rule. When you see a composite function f(g(x)) multiplied by g'(x), you substitute u = g(x) and du = g'(x)dx. This simplifies the integral to ∫ f(u) du, which is often easier to compute. After integrating, substitute back to the original variable.
Integration by parts uses the formula ∫ u dv = uv - ∫ v du. It's useful for products of functions like x·eˣ, x·sin(x), or x·ln(x). Use the LIATE rule to choose u: prioritize Logarithmic, then Inverse trig, Algebraic, Trigonometric, and finally Exponential.
Not all functions have elementary antiderivatives expressible in terms of standard functions. Famous examples include e^(-x²) (the Gaussian integral), sin(x)/x, and x^x. However, definite integrals of these functions can always be computed numerically using methods like Simpson's rule.
The definite integral ∫ from a to b of f(x) dx represents the net signed area between the curve y = f(x) and the x-axis from x = a to x = b. Areas above the x-axis count as positive, and areas below count as negative. The total area (ignoring sign) requires integrating |f(x)|.
Numerical integration approximates definite integrals when symbolic methods fail. This calculator uses adaptive Simpson's rule, which divides the interval into subintervals and approximates each with parabolas. The adaptive version automatically refines regions where the function changes rapidly, achieving high accuracy (error tolerance of 10⁻¹⁰ for regular integrals). For improper integrals with endpoint singularities, precision is reduced but convergent results are still reliable to several decimal places.
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