The given integral is:

$\int \frac{5 \ln(x)}{x(7 + (\ln(x))^2)} \, dx$

Let's solve this step by step.

Step 1: Substitution

We will use the substitution: $u = \ln(x)$ Thus, we have: $du = \frac{1}{x} \, dx$ This substitution simplifies the integral significantly.

Step 2: Rewriting the Integral

Substituting $u = \ln(x)$ and $\frac{1}{x} dx = du$, the integral becomes:

$\int \frac{5u}{7 + u^2} \, du$

Step 3: Solving the Integral

This is a standard integral. The integral of $\frac{u}{7 + u^2}$ is solvable using the substitution $v = 7 + u^2$, but there's a quicker method based on recognizing the derivative of $7 + u^2$. The solution is:

$\int \frac{u}{7 + u^2} \, du = \frac{1}{2} \ln(7 + u^2)$

Therefore, the integral becomes:

$5 \cdot \frac{1}{2} \ln(7 + u^2) = \frac{5}{2} \ln(7 + u^2)$

Step 4: Substituting Back

Now, substitute $u = \ln(x)$ back into the expression:

$\frac{5}{2} \ln(7 + (\ln(x))^2)$

Final Answer:

The solution to the integral is:

$\frac{5}{2} \ln(7 + (\ln(x))^2) + C$

where $C$ is the constant of integration.

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Would you like more details on any step or further clarification?

Related questions:

1. How does substitution help simplify integrals?
2. What are the standard techniques to solve integrals involving logarithms?
3. Can integrals with logarithmic terms always be simplified using substitution?
4. What is the method of partial fractions and when is it applicable to integrals?
5. How can you recognize when an integral has a standard solution form?

Tip:

Always check if a substitution simplifies the integral before attempting more complex methods like integration by parts or partial fractions.