Solution assignment 05 Improper integrals

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Assignment 5




It is clear that this integral converges, because the function |xe^{-x}| is greater than |e^{-x}\sin(x)| from some value of x. The integral of the first function yields a finite result (converges) and this will be the case with the second function. See the graph of e^{-x}\sin(x) in the figure.


The integral has a finite value.

We first look at the calculation of the indefinite integral and use integration by parts:




It seems that this has not helped us a lot, but by applying integration by parts again we get the following result:


Verify this result by differentiation.

Now we get:


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