# 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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