Integration by Parts: Example 3
Integration by Parts: Example 3
Find
The two functions involved in this example do not exhibit any special behavior when it comes to differentiating or integrating. Therefore, we choose one function to be differentiated and the other one to be integrated. We have
which implies
The integration by parts formula gives
The new integral 
is similar in
nature to the initial one. One of the common mistake is to do another
integration by parts in which we integrate 
and differentiate
. This will simply take you back to your original integral
with nothing done. In fact, what you would have done is simply the
reverse path of the integration by parts (Do the calculations to
convince yourself). Therefore we continue doing another integration
by parts as
which implies
Hence
Combining both formulas we get
Easy calculations give
After two integration by parts, we get an integral identical to the
initial one. You may wonder why and simply because the derivative and
integration of 
are the same while you need two derivatives of
the cosine function to generate the same function. Finally easy
algebraic manipulation gives
Try to find out how did we get the constant C?
In fact we have two general formulas for these kind of integrals
and
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