Indefinite Integral
Find antiderivatives and visualize accumulated integrals of any function. Get symbolic antiderivative hints and numerical integration plots with our free indefinite integral calculator.
About This Calculator
This Indefinite Integral Calculator helps students, educators, and professionals find antiderivatives of mathematical functions quickly and visually. It recognizes common function patterns to provide symbolic antiderivative hints and uses numerical integration (trapezoidal rule) to compute and plot the accumulated integral from -5 to 5. You can toggle between viewing the function alongside its integral or exploring the family of antiderivatives shifted by different constants of integration.
Enter any expression using standard mathematical notation: polynomials like x^2 or x^3 - 2x, trigonometric functions like sin(x) or cos(2x), exponential functions like e^x or 2^x, and logarithmic functions like ln(x) or log(x^2 + 1). The antiderivative hint identifies patterns for standard forms, including power rule, trigonometric integrals, exponential integrals, and the reciprocal function.
The chart displays f(x) in blue, the accumulated integral F(x) from -5 to x in orange, and in family mode shows F(x) shifted by C = -2, 0, and +2 to illustrate the geometric meaning of the constant of integration. This visual approach helps bridge the gap between symbolic integration and the Fundamental Theorem of Calculus.
India (IN): Indefinite integrals are a key part of CBSE Class 12 mathematics and JEE Advanced preparation. Students learn integration as the inverse of differentiation, standard integrals, integration by substitution, integration by parts, and partial fractions.
United States (US): Indefinite integrals are covered in AP Calculus AB and BC. Students learn that antiderivatives reverse differentiation, practice finding basic antiderivatives, and apply the constant of integration C in differential equations and initial value problems.
United Kingdom (UK): A-Level Mathematics covers indefinite integration in Pure Mathematics. Students learn to find antiderivatives of standard functions, understand the constant of integration, and apply integration techniques to solve differential equations and find areas under curves.
Frequently Asked Questions
What is an indefinite integral?
An indefinite integral integral f(x) dx represents the family of all antiderivatives of f(x). It is written as F(x) + C, where F'(x) = f(x) and C is the constant of integration (any real number). Unlike a definite integral, it does not have bounds and does not give a numerical value.
How does the calculator find antiderivatives?
The calculator recognizes common patterns in the function expression to suggest the antiderivative form. For simple polynomials, trig functions, and exponentials, it applies standard integration rules. For the accumulated integral visualization, it numerically integrates from -5 to 5 using the trapezoidal rule.
What does the constant of integration C mean?
The constant of integration C represents that antiderivatives are not unique -- any vertical shift of F(x) is also an antiderivative of f(x) since the derivative of a constant is zero. For example, integral 2x dx = x^2 + C because d/dx(x^2) = 2x and d/dx(x^2 + 5) = 2x as well.
What mathematical expressions are supported?
Supported operations: +, -, *, /, ^ (exponentiation), sin(), cos(), tan(), asin(), acos(), atan(), sqrt(), abs(), log()/ln(), exp(), and constants pi and e. The antiderivative hint works best for standard forms like x^n, sin(x), cos(x), e^x, and 1/x.
What is the relationship between indefinite and definite integrals?
The Fundamental Theorem of Calculus connects them: if F is any antiderivative of f, then integralᵇₐ f(x) dx = F(b) - F(a). This means the indefinite integral gives us the tool to evaluate definite integrals. The visualization shows the accumulated integral F(x) = integralˣ₋5 f(t) dt, which is a specific antiderivative with F(-5) = 0.
What are the basic antiderivative formulas?
Common antiderivatives: integral xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), integral 1/x dx = ln|x| + C, integral eˣ dx = eˣ + C, integral aˣ dx = aˣ/ln(a) + C, integral sin(x) dx = -cos(x) + C, integral cos(x) dx = sin(x) + C, integral sec²(x) dx = tan(x) + C, and integral 1/(1+x²) dx = arctan(x) + C.
What is integration by substitution?
Integration by substitution (u-substitution) is a technique for finding antiderivatives of composite functions. If u = g(x) and du = g'(x) dx, then integral f(g(x)) g'(x) dx = integral f(u) du. This reverses the chain rule of differentiation and is especially useful for integrals of the form integral f(ax + b) dx.
What is integration by parts?
Integration by parts is derived from the product rule for derivatives: integral u dv = uv - integral v du. Choose u as the part that simplifies when differentiated and dv as the part that remains manageable when integrated. Common choices include polynomial times exponential, logarithmic times polynomial, or trigonometric times exponential.