https://file.aiquickdraw.com/mathbot/file/172558792542604ocwm8g.jpg

The equation in the image appears to be:

\[
\int \frac{dy}{y(x^2 + y^2)} = \int \frac{dx}{x}
\]

### Solution:

We will solve both integrals separately.

#### Right-hand side: \( \int \frac{dx}{x} \)
This is a standard integral:
\[
\int \frac{dx}{x} = \ln|x| + C_1
\]

#### Left-hand side: \( \int \frac{dy}{y(x^2 + y^2)} \)

To solve this, we can attempt a substitution or recognize a standard form. This is typically solved using the inverse tangent (arctangent) integral, as it resembles the form \(\frac{1}{x^2 + a^2}\). Here, we should consider if polar coordinates are useful, as this resembles a polar integral.

We'll first attempt a standard approach. Let me compute this integral next.The left-hand side integral evaluates to:

\[
\int \frac{dy}{y(x^2 + y^2)} = \frac{\ln|y|}{x^2} - \frac{\ln(x^2 + y^2)}{2x^2}
\]

#### Combining the results:

Equating both sides of the original equation, we have:

\[
\frac{\ln|y|}{x^2} - \frac{\ln(x^2 + y^2)}{2x^2} = \ln|x| + C
\]

This is the general solution for the given integral equation.

### Do you have any questions or need further clarification?

Here are some related questions you might explore:
1. How can we solve similar integrals using substitution techniques?
2. What is the geometric interpretation of the integrals in polar coordinates?
3. How does the logarithmic form arise from integration of rational functions?
4. How do constants of integration affect the general solution of differential equations?
5. Can we solve the equation using polar coordinates?

**Tip**: Always check if a given integral resembles a known standard form, as this can simplify the process significantly.

Solve this: ∫(dy / y(x² + y²)) = ∫(dx / x)

This solution walks through solving the integral equation ∫(dy / y(x² + y²)) = ∫(dx / x). Learn about logarithmic integration and how the left-hand side integral evaluates using rational functions. Suitable for Grades 11-12 and university-level calculus students.

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