Your idea is correct! But you got the integra wrong for v'(z) (you forgot a minus sign!). The correct one is:`v(z)=-10^(-z)/ln(10)`
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) -...
Your idea is correct! But you got the integra wrong for v'(z) (you forgot a minus sign!). The correct one is:
`v(z)=-10^(-z)/ln(10)`
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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