Let's go over each of the questions.

35. Which is the correct substitution for solving the integral:

$\int \frac{1}{x \ln(x)} \, dx ?$

• The goal here is to choose a substitution for $u$ such that the integral becomes easier to solve. The structure suggests that we want $u$ to be something that simplifies the $\ln(x)$ term in the denominator.

• Choice a: $u = \ln(x)$ looks promising because the derivative of $\ln(x)$ is $\frac{1}{x}$, which can help cancel out the $\frac{1}{x}$ factor in the integrand. Let's verify:

  • If $u = \ln(x)$, then $du = \frac{1}{x} dx$.
  • This substitution reduces the integral to: $\int \frac{1}{u} \, du = \ln|u| + C = \ln|\ln(x)| + C$
  • This matches the expected form, so the correct answer is (a).

36. What is the integral of $\ln(x) dx$?

To solve $\int \ln(x) \, dx$, we use integration by parts, where we let:

• $u = \ln(x)$ and $dv = dx$.
• Then $du = \frac{1}{x} dx$ and $v = x$.

Using the integration by parts formula: $\int u \, dv = uv - \int v \, du$ We get: $\int \ln(x) \, dx = x \ln(x) - \int x \frac{1}{x} \, dx = x \ln(x) - x + C$ Thus, the correct answer is: $\boxed{a) \, x \ln(x) - x + C}$

Would you like further clarification on any of these steps? Let me know if you'd like to dive deeper into any of the details.

Here are 5 related questions to consider:

1. How do you perform integration by parts for other functions like $\int x \ln(x) \, dx$?
2. What are the key steps for solving integrals involving natural logarithms?
3. How do you know when to use substitution versus integration by parts?
4. What are some typical substitution strategies for integrals involving logarithmic terms?
5. Can you derive the integral of $\frac{1}{x^2 + 1} \, dx$ using substitution?

Tip:

When solving integrals, identifying patterns that match standard forms (like $\frac{1}{x}$, $\ln(x)$, or powers of $x$) will help you choose the most efficient method, whether it's substitution, parts, or a table of integrals.