https://file.aiquickdraw.com/mathbot/file/1726563567375od9a4jgc.jpg

The integral you've provided is:

\[
\int \frac{\tan(\ln y)}{y} \, dy
\]

To solve this, let's perform a substitution. We will use the following:

- Let \( u = \ln y \). Therefore, \( du = \frac{1}{y} dy \).

This simplifies the integral to:

\[
\int \tan(u) \, du
\]

The integral of \( \tan(u) \) is a standard result:

\[
\int \tan(u) \, du = \ln|\sec(u)| + C
\]

Now, substitute back \( u = \ln y \):

\[
\ln|\sec(\ln y)| + C
\]

So, the solution to the integral is:

\[
\ln|\sec(\ln y)| + C
\]

Would you like further details or have any other questions?

### Related Questions:
1. What is the integral of \( \sec(x) \)?
2. How can we handle integrals involving natural logarithms differently?
3. What is the general approach to solving integrals with trigonometric functions of logarithms?
4. How can substitution be used effectively in solving integrals?
5. Can the result \( \ln|\sec(\ln y)| \) be simplified or expressed in another form?

### Tip:
Substitution is a powerful tool for simplifying complex integrals. When a function's derivative appears in the integrand, substitution can often reduce the problem to a simpler standard integral.

Learn how to solve the integral ∫ (tan(ln y) / y) dy step by step using the substitution method. This example explores the use of logarithmic and trigonometric functions in integration, suitable for advanced calculus students at the university level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation