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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Author: Mohamed Amine Khamsi

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