Answer: \(a=\frac58\).

Set up the integral using the correct limits or antiderivative first. When definite integration is used, evaluate the upper limit and lower limit separately before subtracting.

First integrate:

\(\int \frac{8}{4x+3}\,dx=2\ln(4x+3)+C\).

So

\(\int_0^{2a+1}\frac{8}{4x+3}\,dx=2\ln(8a+7)-2\ln3\).

Using the logarithm law,

\(2\ln\left(\frac{8a+7}{3}\right)=\ln16\).

Since \(\ln16=2\ln4\),

\(\ln\left(\frac{8a+7}{3}\right)=\ln4\).

Therefore

\(\frac{8a+7}{3}=4\), so \(8a+7=12\), and \(a=\frac58\).

This gives the required answer: \(a=\frac58\).