Indefinite Integration by exchange of variables - Problem 2

[Asphyxiated]

Homework Statement

Find:

$$\int \frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx$$

Homework Equations

Same as before:

$$\int f(u)du = F(g(x)) + C$$

The Attempt at a Solution

Let

$$u = 1 + \frac {1}{x}$$

then

$$du = -\frac{1}{x^{2}}$$

so

$$\frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx = \frac {1}{u^{2}} du$$

How does this work out? If u = (1 + 1/x) where did the x^2 go from dx to du?

Then it goes like this:

$$\int \frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx = \int \frac {1}{u^{2}}du = \frac {u^{-1}}{-1} + C = - \frac {1}{1+\frac{1}{x}} + C$$

According to the book this problem can also be done without substituting anything like so:

$$\int \frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx = \int \frac {1}{(1+\frac{1}{x})^{2}} * d(1 + \frac{1}{x}) = \frac {(1 + \frac {1}{x})^{-1}}{-1} + C$$

So my problem here is that I do not understand exactly what is happening, is the reason that we choose u = 1 + 1/x because its derivative is -1/x^2? and that's where the x^2 went in the very first step? Am I on the right track?

 

[zachzach]

Homework Statement

Find:

$$\int \frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx$$

Homework Equations

Same as before:

$$\int f(u)du = F(g(x)) + C$$

The Attempt at a Solution

Let

$$u = 1 + \frac {1}{x}$$

then

$$du = -\frac{1}{x^{2}}$$

so

$$\frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx = \frac {1}{u^{2}} du$$

How does this work out? If u = (1 + 1/x) where did the x^2 go from dx to du?

Then it goes like this:

$$\int \frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx = \int \frac {1}{u^{2}}du = \frac {u^{-1}}{-1} + C = - \frac {1}{1+\frac{1}{x}} + C$$

According to the book this problem can also be done without substituting anything like so:

$$\int \frac {-1}{x^{2}(1+\frac{1}{x})^{2}} dx = \int \frac {1}{(1+\frac{1}{x})^{2}} * d(1 + \frac{1}{x}) = \frac {(1 + \frac {1}{x})^{-1}}{-1} + C$$

So my problem here is that I do not understand exactly what is happening, is the reason that we choose u = 1 + 1/x because its derivative is -1/x^2? and that's where the x^2 went in the very first step? Am I on the right track?

Yes the derivative worked out to allow the x^2 to go away. Maybe you will see it better if you do it like the last one so:


$$du = -\frac{1}{x^{2}}dx$$

or

$$dx = -x^2du$$

So if you plug dx into the integral the negatives and the x^2's cancel. That is why choosing your u to be that is so valuable and that is what you want to generally try to do in u substitution.

 

[Legendre]

Adding on to the other poster...

I prefer not to suppress the denominator.

So, u = 1 + 1/x

du/dx = (-1/x^2) dx/dx, since dx/dx = 1 you can multiply the rhs by it.

So, (-x^2) du/dx = dx/dx.

The dx in your integral can be seen as dx/dx. Substitute it with this value and we are done.

Basically the same thing as the previous poster has said.