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Indefinite Integral

Definition

The set of all antiderivatives of a function ff is the indefinite integral of ff with respect to xx and denoted by ∫f(x) dx.\displaystyle \int f(x)\,\mathrm{d}x\text{.} If FF is an antiderivative of f,f\text{,} then we have ∫f(x) dx=F(x)+C\displaystyle \int f(x)\,\mathrm{d}x=F(x)+C where CC is any constant.

Interpretation

The indefinite integral collects all antiderivatives of ff in one expression. It is not a number but a family of functions, and on an interval the family is F+CF+C: one antiderivative shifted by every constant CC.

The indefinite integral ∫f(x) dx=F(x)+C\int f(x)\,\mathrm{d}x=F(x)+C is the whole family drawn here; the constant CC picks one member, and differentiating any member gives back ff.

The notation deliberately resembles the definite integral. By the fundamental theorem of calculus, any member of the family evaluates definite integrals, ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,\mathrm{d}x=F(b)-F(a), and the constant CC cancels in the difference. An equation such as ∫2x dx=x2+C\int 2x\,\mathrm{d}x=x^2+C is an equality of families, and it is checked by differentiating the right-hand side.

Because the answer is a family, correct answers can look different. Both sin⁡2x+C\sin^2x+C and −cos⁡2x+C-\cos^2x+C are correct for ∫2sin⁡xcos⁡x dx\int 2\sin x\cos x\,\mathrm{d}x: differentiating either gives 2sin⁡xcos⁡x2\sin x\cos x, and sin⁡2x−(−cos⁡2x)=1\sin^2x-(-\cos^2x)=1 is a constant.

The single constant CC relies on the domain being an interval. For ∫dxx=ln⁡∣x∣+C\int\frac{\mathrm{d}x}{x}=\ln|x|+C the formula holds on (−∞,0)(-\infty,0) and on (0,∞)(0,\infty) separately: the function equal to ln⁡∣x∣\ln|x| for x<0x<0 and to ln⁡∣x∣+1\ln|x|+1 for x>0x>0 also has derivative 1/x1/x at every x≠0x\ne0.

Later, the rules of substitution and integration by parts help to find antiderivatives that cannot be read off directly.

Key Results

  • Indefinite integrals are defined up to an additive constant.
  • Differentiation recovers the original integrand from any antiderivative.

Examples

  • ∫2x dx=x2+C.\displaystyle\int 2x\,\mathrm{d}x = x^2 + C\text{.}
  • ∫cos⁡x dx=sin⁡x+C.\displaystyle \int \cos x\,\mathrm{d}x = \sin x + C\text{.}

Prerequisites

Sources

Symbols: infinity, integral

Last updated: 20 October 2026 03:02