Is this an exact differential or not? Apply the test.
which do not agree. Therefore, it is not an exact differential, and the two functions and have not come from a common original function.
It is possible in such cases to discover, however, an integrating factor, that is to say, a factor such that if both are multiplied by this factor, the expression will become an exact differential. There is no one rule for discovering such an integrating factor; but experience will usually suggest one. In the present instance will act as such. Multiplying by , we get
.
Now apply the test to this.
which agrees. Hence this is an exact differential, and may be integrated. Now, if ,