Choosing a substitution for ∫(x cos(x) − sin(x))/(x − sin(x))² dx

[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?

 

[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.

 

[dirk_mec1]

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

 

[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.

 

[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.

 

[dirk_mec1]

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.

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

 

[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.