I'm trying to solve this physics equation: $$E=\frac {\lambda y} {4\pi \epsilon_{0}} \int_{-\frac l 2}^\frac l 2 \frac {dx} {({x^2+y^2})^\frac 3 2} $$
However, my calculus is a little rusty and am told I need to do a trig substitution. My question is, when do we use trig substitution?
Letting $x=y\tan\theta,$ $$\frac{dx}{d\theta}=\frac{y}{\cos^2\theta}.$$ Hence, your definite integral will be $$\begin{align}\int_{\alpha}^{\beta}\frac{1}{(y^2\tan^2\theta+y^2)^{3/2}}\cdot\frac{y}{\cos^2\theta}d\theta&=\int_{\alpha}^{\beta}\frac{1}{(y/\cos\theta)^3}\cdot\frac{y}{\cos^2\theta}d\theta\\&=\int_{\alpha}^{\beta}\frac{\cos\theta}{y^2}d\theta\\&=\frac{1}{y^2}[\sin\theta]_{\alpha}^{\beta}\\&=\frac{\sin\beta-\sin\alpha}{y^2}\end{align}$$ where $$-\frac{l}{2}=y\tan\alpha,\ \ \ \frac{1}{2}=y\tan\beta.$$ Here, I used $$1+\tan^2\theta=\frac{1}{\cos^2\theta}.$$