Homework Help: Easy integral

1. Jul 6, 2008

dirk_mec1

$$\int \frac{x \cos(x) - \sin(x)}{( x-\sin(x))^2}\ \mbox{d}x$$

I don't see which substitution I should use can anyone help me?

2. Jul 6, 2008

morphism

The integrand looks a lot like something you'd get if you differentiated an expression of the form f(x)/g(x) using the quotient rule.

3. Jul 7, 2008

dirk_mec1

That I know, but it isn't helping me. I'm interested in a structured manner of solving this problem.

4. Jul 7, 2008

matt grime

So guessing a substitution is structured, but noting the form of the integrand is 'unstructured'? That seems like a highly arbitrary choice to make. Since integrals are generically impossible to do by hand, I'd take what you can get when you can get it.

5. Jul 7, 2008

TheoMcCloskey

Indeed, with morphism's observation, a keen eye, and some algebra, you can easily arrive at the solution without any significant integration. But I have no other useful suggestion if your intent is otherwise.

6. Jul 7, 2008

dirk_mec1

Fine. So how do I proceed next? I'm seeing something of a quotient rule but how can you find the primitive?

7. Jul 7, 2008

Dick

Just guess. Write down the form of (f(x)/g(x))'. What's a good guess for g(x)? Put that in. That's easy. Now start hunting for an f(x) that works.