How to integrate x^2 / (xsin(x) + cos(x))

[harimakenji]

Homework Statement

int [x² / (x sin x + cos x)²]dx

Homework Equations

integration

The Attempt at a Solution

can anyone help me to start?

thank you very much

 

[MathematicalPhysicist]

Hint:(cos(x)/x)'= (-xsin(x)- cos(x))/x^2

I am not sure it helps though.

 

[fzero]

Note that

$$d\left(\frac{1}{x \sin x+\cos x} \right) = - \frac{x \cos x \, dx}{(x \sin x+\cos x)^2}$$

and use integration by parts.

 

[harimakenji]

Note that

$$d\left(\frac{1}{x \sin x+\cos x} \right) = - \frac{x \cos x \, dx}{(x \sin x+\cos x)^2}$$

and use integration by parts.

I am still not getting your hint. When using integration by parts, I have to choose u and dv. I don't know how to choose u and dv for this question.
How to apply your hint to solve this question?

thank you very much

 

[fzero]

I am still not getting your hint. When using integration by parts, I have to choose u and dv. I don't know how to choose u and dv for this question.
How to apply your hint to solve this question?

thank you very much

Try

$$v = \frac{1}{x \sin x+\cos x} .$$

u will be whatever is left over in the original integrand after rewriting it in terms of dv. u will not look like something nice to integrate, but v du is fairly simple.

 

[harimakenji]

Try

$$v = \frac{1}{x \sin x+\cos x} .$$

u will be whatever is left over in the original integrand after rewriting it in terms of dv. u will not look like something nice to integrate, but v du is fairly simple.

I got it. I have tried u = (x / cos x) before and was really unsure that this was the right method, but as you said, it really became a simple integration. I gave up before trying to find the term v du, can't let it happen anymore.

Thank you very much for your hint and help. I really appreciate it.