I have the following exepression in my book:

$\frac{dx}{dt}+{a}_{1}(t)x=g(t),\text{}\text{}\text{}\text{}x({t}_{0})={x}_{0}$

Then it says, multiply both sides of the differential equation by the integrating factor $I(t)$.

$I(t)\frac{dx(t)}{dt}+{a}_{1}(t)I(t)x(t)=I(t)g(t)$

So far so good. Hereafter it says, the left-hand side is an exact derivative.

$\frac{d[x(t)I(t)]}{dt}=I(t)g(t)$

And my question is, how does the book come to the last? Can anyone give a HINT.

$\frac{dx}{dt}+{a}_{1}(t)x=g(t),\text{}\text{}\text{}\text{}x({t}_{0})={x}_{0}$

Then it says, multiply both sides of the differential equation by the integrating factor $I(t)$.

$I(t)\frac{dx(t)}{dt}+{a}_{1}(t)I(t)x(t)=I(t)g(t)$

So far so good. Hereafter it says, the left-hand side is an exact derivative.

$\frac{d[x(t)I(t)]}{dt}=I(t)g(t)$

And my question is, how does the book come to the last? Can anyone give a HINT.