Integration of quotient of derivative of function by function

Statement

Version for indefinite integration

We have the following rule for indefinite integration for a differentiable function $f$ on an interval:

$\int {\frac {f'(x)}{f(x)}}\,dx=\ln |f(x)|+C$

This indefinite integral makes sense over any interval such that $f(x)$ is nonzero everywhere on the interval. If we are trying to find an antiderivative over a union of different intervals (which are not connected) then the value of the constant $C$ may be different in the different intervals.

Further note: Over any interval where $f$ is differentiable and nowhere zero, it is either uniformly positive or uniformly negative. Thus, the correct antiderivative is either $\ln(f(x))+C$ or $\ln(-f(x))+C$, and we may be able to use additional information to figure out which one it is.

Version for definite integration

Consider a function $f$ and a closed interval $[a,b]$ in the domain of $f$ such that:

• $f$ is a continuous function on a closed interval $[a,b]$.
• $f$ is a differentiable function on the open interval $(a,b)$.
• $f$ does not take the value zero at any point on $[a,b]$.

Then, we have the following rule for definite integration:

$\!\int _{a}^{b}{\frac {f'(x)}{f(x)}}\,dx=\ln \left({\frac {f(b)}{f(a)}}\right)$

Note that we don't care about the differentiability of $f$ at the boundary points $a,b$. If $f$ is not differentiable at these points, the definite integral is an improper integral, but the formula still holds.

Note also that we do not need an absolute value symbol here, because both $f(a)$ and $f(b)$ have the same sign, so the quotient is automatically positive.

Examples

Examples of indefinite integrals

• Tangent function
• Cotangent function