Hi guys is there anyone who is able to explain me in more detail why statements in this screenshot hold? Which rule did the author exactly use? enter image description here
$${\text{To be precise, let $b(t)$ be a deterministic differentiable function with $b(0)=0$.}\\~\\\text{We know that:}\\~\\\qquad\dfrac 12\dfrac{\mathrm d b^2(s)}{\mathrm d s} = b(s)\,\dfrac{\mathrm d b(s)}{\mathrm d s}\\~\\\text{Integrating both sides, we obtain by the classical rules of calculus.}\\~\\\qquad\begin{align}\dfrac 12\int_0^t\dfrac{\mathrm db^2(s)}{\mathrm d s}\,\mathrm d s &=\int_0^t b(s)\dfrac{\mathrm d b(s)}{\mathrm d s}\,\mathrm d s\\\dfrac 12b^2(t)&=\int_{0}^{b(t)} b(s)\,\mathrm d b(s)\end{align}}$$
The first line is the chain rule. Differentiating $b^2$ with respect to $s$ gives $2b \frac{db}{ds}$.
In the second line, we have:
$\int \frac{d b^2}{ds} ds = b^2$ is the Fundamental Theorem of Calculus - integrating and differentiating are inverse operations.
$\int b \frac{db}{ds} ds = \int b\ db$ is essentially a substitution that puts $b = b(s)$ into the integral.
And the middle $=$ sign is because we've integrated both sides of the first line consistently.