91Ó°ÊÓ

First, we need to choose and define the functions u and dv. We can choose: $$u = \ln(2x+3)$$ and $$dv = dx$$

Now we'll find the derivatives and antiderivatives of u and dv respectively. To find du, differentiate u with respect to x: $$du = \frac{2}{2x+3} dx$$ To find v, integrate dv with respect to x. Since we are given the condition to introduce a constant of integration $$C_{1} = \frac{3}{2}$$, we will integrate dv in the following way: $$v = \int dx + C_{1} = x + \frac{3}{2}$$

Now we'll apply the integration by parts formula: $$\int u dv = uv - \int v du$$ Substitute the expressions for u, dv, du, and v: $$\int \ln(2x+3) dx = (\ln(2x+3))(x+\frac{3}{2}) - \int (x+\frac{3}{2})\frac{2}{2x+3} dx$$

Now we'll compute the integral: First, let's split the integral: $$\int (x+\frac{3}{2})\frac{2}{2x+3} dx = \int x\frac{2}{2x+3} dx + \int \frac{3}{2}\frac{2}{2x+3} dx$$ Now, observe the second integral: $$ \int \frac{3}{2}\frac{2}{2x+3} dx = \frac{3}{1}\int\frac{1}{2x+3} dx $$ We can substitute $$t=2x+3$$, hence $$dt = 2 dx$$ and $$\frac{1}{2} dt = dx$$ Now our integral is easier to compute: $$ \frac{3}{1}\int\frac{1}{t} \frac{1}{2} dt = \frac{3}{2}\int\frac{1}{t} dt $$ Now, the first integral $$\int x\frac{2}{2x+3} dx$$ can be computed using substitution or observation: Let $$z=2x+3$$, then $$dz = 2dx.$$ Rewriting the integral in terms of z, we get: $$\frac{1}{2}\int\frac{z-3}{z}dz$$ Now, split the integral into two: $$\frac{1}{2}\left[\int\frac{z}{z}dz - \int\frac{3}{z}dz\right]$$ The first integral results in just z and the second integral can be computed as the natural logarithm of the absolute value of z multiplied by the constant. So, we get: $$\frac{1}{2}\left[z-3\ln\lvert z\rvert + C\right]$$ Now, substitute z back with $$2x+3$$: $$\frac{1}{2}\left[(2x+3)-3\ln\lvert 2x+3\rvert + C\right]$$ Finally, combine the two integrals: $$\int \ln(2x+3) dx = (\ln(2x+3))(x+\frac{3}{2}) - \left[\frac{1}{2}\left[(2x+3)-3\ln\lvert 2x+3\rvert + C\right] + \frac{3}{2}\ln\lvert 2x+3\rvert\right] + C_{2}$$ Where $$C_{2}$$ is the constant of integration.