Wednesday, June 8, 2016

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....

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.


` `


` `

No comments:

Post a Comment

Is Charlotte Bronte's Jane Eyre a feminist novel?

Feminism advocates that social, political, and all other rights should be equal between men and women. Bronte's Jane Eyre discusses many...