# Confused by this integration

1. Oct 15, 2015

### Abdul.119

So I was reading a solution of a problem that has to do with integration in polar coordinates, and in one of the steps it did this

How did these terms (circled in blue) change like that? I've never seen a step like that before..

2. Oct 15, 2015

### Staff: Mentor

What is the differential of $\sin \theta$?

If y = f(x), the differential of y is defined as dy = f'(x) dx

3. Oct 15, 2015

### pwsnafu

The substitution rule says
$\int f(x) \, \frac{du}{dx} \, dx = \int f(u(x)) \, du(x)$
Hence, $\theta = x$, $\frac{du}{dx} = \cos\theta$ and $u = \sin \theta$ gives
$\int f(\theta) \, cos(\theta) \, d\theta = \int f(\theta) \, d(\sin(\theta))$

4. Oct 15, 2015

### Abdul.119

Why didn't they use the regular u-substitution, where u = cos(θ) , and du = -sin(θ) dθ ?

5. Oct 15, 2015

### mathman

u = sin(θ) , and du = cos(θ) dθ avoids the complication of sign change.

6. Oct 16, 2015

### HallsofIvy

Staff Emeritus
I'm not sure why you call that "the regular u-substitution". To integrate something like $\int sin(\theta) cos(\theta) d\theta$ either $u= cos(\theta)$, $du= -sin(\theta)d\theta$ or $u= sin(\theta)$, $du= cos(\theta)d\theta$ will work. And, I suspect that most people would use the latter since, as mathman said, it avoids the negative sign.