Integration of quotient of derivative of function by function
Version for indefinite integration
This indefinite integral makes sense over any interval such that 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 may be different in the different intervals.
Further note: Over any interval where is differentiable and nowhere zero, it is either uniformly positive or uniformly negative. Thus, the correct antiderivative is either or , and we may be able to use additional information to figure out which one it is.
Version for definite integration
Consider a function and a closed interval in the domain of such that:
- is a continuous function on a closed interval .
- is a differentiable function on the open interval .
- does not take the value zero at any point on .
Then, we have the following rule for definite integration:
Note that we don't care about the differentiability of at the boundary points . If 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 and have the same sign, so the quotient is automatically positive.