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Reciprocal Rule of Integration


$\displaystyle \int{\dfrac{1}{x} \,} dx \,=\, \ln{x}+c$

The integral of multiplicative inverse of a variable equals to sum of natural logarithm of variable and constant of integration is called the reciprocal rule of integration.


Let $x$ be a variable. The multiplicative inverse or reciprocal of variable $x$ is expressed as $\dfrac{1}{x}$ in mathematical form. It is often appeared in integral calculus. Hence, an integral rule with reciprocal of a variable is introduction in integration.

The integration of multiplicative inverse of $x$ with respect to $x$ is written in mathematical form as follows.

$\displaystyle \int{\dfrac{1}{x} \,} dx$

The integration of reciprocal of $x$ with respect to $x$ is equal to natural logarithm of variable $x$ and integral constant. It is mathematically expressed in the following form in calculus.

$\implies$ $\displaystyle \int{\dfrac{1}{x} \,} dx \,=\, \ln{(x)}+c$

This integral rule is used as a formula in calculus when a variable is involved in integration reciprocally.

Alternative forms

The multiplicative inverse rule of integration can be written in terms of any variable in calculus.

$(1) \,\,\,$ $\displaystyle \int{\dfrac{1}{m} \,} dm \,=\, \ln{(m)}+c$

$(2) \,\,\,$ $\displaystyle \int{\dfrac{1}{y} \,} dy \,=\, \ln{(y)}+c$


Learn how to prove the integration of multiplicative inverse of a variable in integral calculus.

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