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Integral of a Derivative

Description

This equation is part of the fundamental theorem of calculus, and shows that the integral of a derivative is the original function.

\[\htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000065}{f'(x)} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) + \htmlClass{sdt-0000000070}{C}\]

Symbols Used:

This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc.

\( \int \)

This is the symbol for an integral, sometimes referred to as an antiderivative. Graphically, it can be understood as the area between a curve and the axis the integral is taken with respect to.

\( f'(x) \)

This is a general symbol for the derivative of a general function, \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})\), in Legrange's notation.

\( C \)

This symbol represents the constant of integration. It must be added to the result of all definite integrals to encompass all possible solutions that satisfy the integral.

\( f \)

This is the symbol for a function. It is commonly used in algebra, and (multivariate) calculus.

\( \:d \)

This is the symbol for a differential. It represents an infinitesimally small (infinitely close to zero) change in whatever variable it is with respect to.

Derivation

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Example

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