Problem 17 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{\ln x}{x(3+2 \ln x)} d x $$

Step-by-Step Solution

Answer

The integral solution is \( \frac{1}{2} (3 + 2 \ln x) - \frac{3}{2} \ln|3+2 \ln x| +C \)

Step 1: Set up the integral

First, we set up the integral to be solved, which is given as \( \int \frac{\ln x}{x(3+2 \ln x)} dx \). Now we need to identify which function should be set as 'u' for substitution.

Step 2: Apply substitution

In this scenario, the denominator of the fraction looks complicated, so let's try substituting it. We can designate \( u = 3 + 2 \ln x \), which means \[ du = \frac{2}{x} dx \] and \( x = e^{(u-3)/2} \] .

Step 3: Convert the integral into 'u' terms

Now we convert everything inside the integral to terms of 'u'. \[ \int \frac{\ln x}{x(3+2 \ln x)} dx = \int \frac{\ln(e^{(u-3)/2})}{e^{(u-3)/2}u} + (e^{(u-3)/2})du \]This simplifies to \[ \int \frac{u-3}{2u} du = \int \frac{1}{2} du - \frac{3}{2u} du \]

Step 4: Integrate.

Now we can integrate term by term to find \[ \frac{1}{2} u - \frac{3}{2} \ln|u| + C \]

Step 5: Substitute 'u' back in.

Now we substitute 'u' back in to our equation to find the solution in terms of 'x'. This gives \[ \frac{1}{2} (3 + 2 \ln x) - \frac{3}{2} \ln|3+2 \ln x| +C \]