Electric Field due to a charged line; With y as function of theta

[Molderish]

Well the main problem here its once a i get the integral i do not how to put dy as function of thetha and then integrate ; at the end that's what i get

Ex=integral (dE*cos$\theta$)=($\lambda$/4$\pi$$\epsilon$)
times the integral of cos($\theta$)dy/x^2+y^2

where i can easily find the cosine and do the integral but I'm asked to express strictly "y" as function of thetha so as cos(θ)=x/√x^2+y^2 so 1/√x^2+y^2=cos^2(θ)/x^2

and i know that y=x*tan(θ) but i don't how to get "dy" graphically nor analytically;

 

[Simon Bridge]

Why not do the integral in terms of dy?

 

[Molderish]

Well the professor asked us for bring both methods in terms of y as function of theta & theta as function of y;

I already did in terms of dy. making the substitution of cosine(θ) = x/√x^2+y^2 and then integrating.

but now i need the other method. y in terms of "θ"

 

[Simon Bridge]

OK - so complete: $$y=x\tan\theta\\ \implies \frac{dy}{d\theta}=\cdots$$ ... then multiply both sides by $d\theta$.

Then substitute these values for y and dy into:
$$E_x=\frac{\lambda}{4\pi\epsilon_0}\int \frac{\cos\theta\; dy}{x^2+y^2}$$ ... with appropriate limits.

note: I wrote all that out so you'd get an example of LaTeX, since I noticed you struggling with typing out the math there.
To figure out how I did it - just hit the "quote" button attached to this post.
The equations all appear between dollar or hash marks.