So, in my book, Circuit Analysis, Second Edition by Cunningham and Stuller the give the following integration for the the energy absorbed by a capacitor in circuit. Remember $p=vi$ (power absorbed = voltage * current) and capacitor voltage is $i_{ab}=C\frac{d}{dt}v_{ab}$, (momentary current is equal to the momentary change in voltage across the terminals $a$ and $b$.) The derevation goes like this: $$w(t)=\int_{-\infty}^tv({\lambda})i(\lambda)\mathrm{d}\lambda\\=\int_{-\infty}^{t}v(\lambda)C\left[\frac{d}{d\lambda}v({\lambda})\right]\mathrm{d}{\lambda}$$
and proceeds to the last step, which I do not understand.
$$\int_{v(-\infty)}^{v(t)}Cv(\lambda)dv({\lambda}) = \frac{1}{2}Cv^2(t)$$
I have no idea what technique is used to do this. In fact, the syntax $\int dx({\lambda})$ is totally unfamiliar to me; I don't even know where to go to look that up. Also, I do not follow the integration ie, how they get the part of the line $\frac{1}{2}Cv^2(t)$. Worse, I'm not totally sure how they changed the bounds of integration. The first two lines are easy to follow.
A thousand thank yous for any guidance I may receive.