# integration

#### paulmdrdo

##### Active member
how would you integrate

1. $\displaystyle \int \left(\frac{x}{y}\right) d\left(\frac{x}{y}\right)$

2. $\displaystyle \int \frac{d(xy)}{(xy)^3}$

if the two is just in this form I could easily answer it

$\displaystyle \int d\left(\frac{x}{y}\right)=(\frac{x}{y})$

$\displaystyle \int d(xy)=(xy)$

but I don't know how to treat $\frac{x}{y}$ in 1 and the $(xy)^3$ in 2.

#### mente oscura

##### Well-known member
how would you integrate

1. $\displaystyle \int \left(\frac{x}{y}\right) d\left(\frac{x}{y}\right)$

2. $\displaystyle \int \frac{d(xy)}{(xy)^3}$

if the two is just in this form I could easily answer it

$\displaystyle \int d\left(\frac{x}{y}\right)=(\frac{x}{y})$

$\displaystyle \int d(xy)=(xy)$

but I don't know how to treat $\frac{x}{y}$ in 1 and the $(xy)^3$ in 2.

Hello.

$$1) \ u=\dfrac{x}{y}$$

$$\displaystyle \int u \ d(u)=u= \dfrac{x}{y}$$

$$2) \ u=(xy)$$

$$\displaystyle \int \dfrac{d(u)}{u^3}=?$$

Can you follow?

Regards.

#### MarkFL

Staff member
Hello.

$$1) \ u=\dfrac{x}{y}$$

$$\displaystyle \int u \ d(u)=u= \dfrac{x}{y}$$
What we want here is:

$$\displaystyle \int u\,du=\frac{1}{2}u^2+C$$

#### mente oscura

##### Well-known member
What we want here is:

$$\displaystyle \int u\,du=\frac{1}{2}u^2+C$$
Right. I I'm wrong.

I wrote:

$$\int u\,du$$

And I thought in:

$$\int \,du$$

I'm sorry.

Regards.