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8 Nov 2024
where is the constant of integration.
In calculus, integrating functions that combine both polynomial and logarithmic terms, like , requires the specific technique of integration by parts. This method is particularly useful for integrating products of different types of functions, such as polynomials and logarithms.
To integrate , we will use the integration by parts formula:
For our integral :
– Let , so
– Let , so
This setup allows us to apply the integration by parts formula effectively. Its important to know which to set as and which as . In general, if it is easy to take the derivative, you should set that as , while if it is easy to integrate, you should set that as .
Now substitute into the formula :
Simplify inside the integral:
Now we integrate :
So our expression becomes:
where is the constant of integration.
To summarize, the integral of is . By using integration by parts, we were able to handle the product of the polynomial and logarithmic terms effectively as long as we correctly identified which should be integrated, and which would be easier differentiated.