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

  • Thread starter Thread starter harimakenji
  • Start date Start date
  • Tags Tags
    Integrate
harimakenji
Messages
89
Reaction score
0

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
 
Physics news on Phys.org


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

I am not sure it helps though.
 


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.
 


fzero said:
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
 


harimakenji said:
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.
 


fzero said:
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.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top