Stuck on this problem. I don't know how they got the substitution for $x = r tan(\theta)$. I know they take the derivative for $x = r tan(\theta)$, but I am not understanding what they took the derivative respect to; changing from $dx \rightarrow d\theta$ for the substitution.
Also, the limits on integration. When I tried to change the limits from $\int_{-\infty}^{\infty}$ to $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}$ I get it diverges. Some clarity in how my math is wrong would be awesome!
According with the triangle formed by $s,r$ and $x$ there are the relations $$s=\frac {r}{cos\theta}$$ $$x=r\tan\theta$$ being $r$ a constant.
When $\theta$ changes so does $x$. So, $dx=r·d(tan\theta)= \frac {r}{cos^2\theta}d\theta$
Substitute in the integral: $$E_y= \int_{x=-\infty}^{x=\infty} k \frac {\lambda dx}{s^2} cos \theta \;=\; \int_{x=-\infty}^{x=\infty} k\lambda \frac {r·d\theta}{cos^2\theta}·\frac{cos^2\theta}{r^2}·cos\theta \;=\; \int_{x=-\infty}^{x=\infty} \frac {k\lambda}{r} cos\theta \;d\theta$$
For the limits of integration, notice that $x=-\infty$ when $\tan\theta=-\infty$ which means $\theta=-\frac{\pi}{2}$ The same way, for $x=\infty$ we get $\theta=\frac{\pi}{2}$
Taking the constants out of the integral and replacing the limits we reach to $$E_y= \int_{x=-\infty}^{x=\infty} k \frac {\lambda dx}{s^2} cos \theta \;=\; \frac {k\lambda}{r} \int_{\theta=-\frac{\pi}{2}}^{\theta=\frac{\pi}{2}} cos\theta \;d\theta$$