Integration by Parts Contradiction

[neo_]

Ok guys, this is my first post. Please go easy...

This question is from Morris Kline's Calculus: An Intuitive and Physical Approach and unfortunately there aren't solutions for all questions (really annoying).

I'm not even sure if this counts as a contradiction but anyway:

Let us evaluate int.(1/x)dx by parts. If we let u=1/x and dv=1dx, we obtain int.(dx/x)=1 + int.(dx/x). Then 1=0. What is wrong?

I would really appreciate a simple explanation from any of you experienced brains out there! Thanks.

 

[VietDao29]

Here's what I think. Let:
$$g(x) := \int f(x)dx \quad \mbox{and} \quad h(x):= \int f(x)dx$$. Then, you do not have g(x) - h(x) = 0, you will have g(x) - h(x) = C, where C is some constant.
So here's the same, you can say that:
$$\int \frac{dx}{x} - \int \frac{dx}{x} = C$$, where C is some constant.
So it's not a contradiction...
Viet Dao,

 

[neo_]

I get you Viet Dao... I don't think I would not have thought that way at all on my own... not tonight anyway. Thanks.

Here's a thought I just had:
Could int.(dx/x) = C + int.(dx/x) , where C is some constant other than 1, be eventuated from Int. by Parts?

 

[VietDao29]

You can continue integrating by parts, something like:
$$\int \frac{dx}{x} = 1 + \int \frac{dx}{x} = 1 + \left( 1 + \int \frac{dx}{x} \right) = 1 + 1 + .. + 1 + \left( 1 + \int \frac{dx}{x} \right)$$.
So you'll have:
$$\Leftrightarrow \int \frac{dx}{x} - \int \frac{dx}{x} = 1 + 1 + 1 + ... + 1$$.
Viet Dao,

 

[neo_]

Gotcha, excellent explanation.