Antiderivative Calculator

What Is an Antiderivative?

An antiderivative of a function f(x) is any function F(x) whose derivative equals f(x). If f(x) = 2x, then F(x) = x² + C is an antiderivative because the derivative of x² + C is 2x. The "C" (constant of integration) represents an entire family of functions that differ only by a vertical shift, since the derivative of any constant is zero. Finding antiderivatives is the central operation of integral calculus and is essential for calculating areas, volumes, accumulated quantities, and solving differential equations. Enter any function in the calculator above for an instant and complete antiderivative computation with detailed step-by-step explanation.

How Is an Antiderivative Different from an Integral?

An antiderivative and an indefinite integral are mathematically the same thing: both find F(x) + C where F'(x) = f(x). The terms are used interchangeably. A definite integral evaluates the antiderivative between two bounds: the integral from a to b of f(x) dx = F(b) - F(a). The antiderivative is the function, the definite integral is a number. This page focuses on finding the antiderivative function. Our Integral Calculator page covers both indefinite and definite integration with applications to area and volume calculations.

Basic Antiderivative Rules

Power rule: The antiderivative of x^n is x^(n+1)/(n+1) + C, for n not equal to -1. The antiderivative of x³ is x⁴/4 + C. The antiderivative of x⁻² is -x⁻¹ + C = -1/x + C. Special case: The antiderivative of 1/x is ln|x| + C. Exponential: The antiderivative of e^x is e^x + C. The antiderivative of a^x is a^x/ln(a) + C. Trigonometric: The antiderivative of cos(x) is sin(x) + C. The antiderivative of sin(x) is -cos(x) + C. The antiderivative of sec²(x) is tan(x) + C.

Why Is Finding Antiderivatives Harder Than Derivatives?

Differentiation follows systematic rules that always produce an answer. Given any elementary function, you can compute its derivative mechanically. Antidifferentiation has no such guarantee. Some simple-looking functions, like e^(-x²) (the bell curve), have no antiderivative expressible in elementary functions. Others, like 1/(1+x³), have antiderivatives involving complex combinations of logarithms and arctangent that are far more complicated than the original function. This asymmetry between derivatives and antiderivatives is a fundamental feature of calculus, not a limitation of any particular technique.

Substitution Method (U-Substitution)

U-substitution reverses the chain rule. For the antiderivative of 2x cos(x²): let u = x², then du = 2x dx. The integral becomes cos(u) du = sin(u) + C = sin(x²) + C. The key is recognizing a composite function where the derivative of the inner function appears as a factor. For the antiderivative of x e^(x²): let u = x², du = 2x dx, so x dx = du/2. The integral becomes (1/2)e^u du = (1/2)e^(x²) + C. U-substitution is the most commonly used integration technique and should be the first method you try when basic rules do not apply directly.

Integration by Parts

Integration by parts reverses the product rule: the integral of u dv = uv minus the integral of v du. Choose u and dv so that the resulting integral is simpler. For the antiderivative of x e^x: let u = x (which simplifies when differentiated) and dv = e^x dx (which stays manageable when integrated). Then du = dx and v = e^x. Result: x e^x minus the integral of e^x dx = x e^x - e^x + C = e^x(x-1) + C. The LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) helps choose u: pick the function earliest in this list.

Antiderivative Table

Common antiderivatives worth memorizing: antiderivative of x^n = x^(n+1)/(n+1) + C. Antiderivative of 1/x = ln|x| + C. Antiderivative of e^x = e^x + C. Antiderivative of sin(x) = -cos(x) + C. Antiderivative of cos(x) = sin(x) + C. Antiderivative of sec²(x) = tan(x) + C. Antiderivative of 1/(1+x²) = arctan(x) + C. Antiderivative of 1/sqrt(1-x²) = arcsin(x) + C. These form the building blocks for all more complex antiderivative computations using substitution, integration by parts, and partial fractions. Having this reference eliminates the need to derive each antiderivative from scratch. With these building blocks memorized, the integration techniques of substitution and integration by parts become straightforward applications of pattern recognition rather than inventive problem-solving.

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