(a) Given that

\(\int_1^a\left(\frac1x-\frac{1}{2x+3}\right)\,dx=\ln3,\)

where \(a\gt 0\), find the exact value of \(a\), giving your answer in simplest surd form.

(b) Find the exact value of

\(\int_0^{\pi/3}\left(\sin\left(2x+\frac{\pi}{3}\right)-1+\cos2x\right)\,dx.\)