$$\begin{gathered}\int_a^b f = F(b)-F(a)\\[2pt] \int_a^c = \int_a^b + \int_b^c, \quad \int_a^a = 0\end{gathered}$$

Constants factor out, sums split, swapping limits flips the sign. Reflex: Split the integrand, chain the ranges, integrate the explicit part yourself.

$$\begin{gathered}\frac{3x+2}{\sqrt{4x-3}}\\[2pt] = \frac{1}{2}\cdot\frac{6x+1}{\sqrt{4x-3}} + \frac{1.5}{\sqrt{4x-3}}\end{gathered}$$

The "show" part hands you an antiderivative — reuse it on the matching chunk.

Reflex: scale to match the leading term, patch the constant — the remainder is a power-rule integral.