Integration by Parts: Example 2
Integration by Parts: Example 2
Evaluate
First let us point out that we
have a definite integral. Therefore the final answer will be a number
not a function of x! Since the derivative or the integral of 
lead to the same function, it will not matter whether we do one
operation or the other. Therefore, we concentrate on the other
function 
. Clearly, if we integrate we will increase the power.
This suggests that we should differentiate and integrate .
Hence
After integration and differentiation, we get
The integration by parts formula gives
It is clear that the new integral 
is not
easily obtainable. Due to its similarity with the initial integral,
we will use integration by parts for a second time. The same
discussion as before leads to
which implies
The integration by parts formula gives
Since 
, we get
which finally implies
Easy calculations give
From this example, try to remember
that most of the time the integration by parts will not be enough to
give you the answer after one shot. You may need to do some extra
work: another integration by parts or use other techniques,....
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