The question is evaluate the integral of : `int`  `z/10^z`  dz using integration by parts.

I have u=z,du=1,v=(-10^z)/ln(10),dv=10^z

I am stuck. Are these correct and how do i move on in the problem after plugging them in because the integral seems extremely complicated. I need help asap because my homework is due tomorrow morning.



Expert Answers

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Your idea is correct! But you got the integra wrong for v'(z) (you forgot a minus sign!). The correct one is:


You can check that this is the correct v(z) by taking its derivative with respect to z. You got it right with u(z) and u'(z), so I will use then for the integration by parts.

Now, following the integration by parts we have:
`intu(z)v'(z)=u(z)v(z) - intu'(z)v(z)`

Inserting our functions:

`intz/10^z = -z10^(-z)/ln(10) - int(-10^(-z)/ln(10))=`

`= -z10^(-z)/ln(10) + int10^(-z)/ln(10)`

All that remains is to evaluate the integral of `10^(-z)/ln(10)`

But we know that integral. It is simply `10^(-z)/(ln^2(10)) + C`

Thus, we get as a final result:

`intz/10^z = -z10^(-z)/ln(10) - 10^(-z)/(ln^2(10)) + D` 

I used D for constant because that may not be the same value as C, due to the integral being indefinite on the first term aswell!

I believe your problem was with the minus sign, which could have resulted in a really complicated integral.

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