Dy/dx=cos5x please proper solution and make sure the answer is correct and teach me what i can write in the paper.

To solve the differential equation 

\[
\frac{dy}{dx} = \cos(5x),
\]

we need to integrate both sides with respect to \(x\). Here's the step-by-step process you can write on your paper:

### Step 1: Write down the integral

Start by integrating both sides of the equation:

\[
y = \int \cos(5x) \, dx.
\]

### Step 2: Use substitution to simplify the integral

Let’s use substitution to simplify the integral. Set:

\[
u = 5x \implies \frac{du}{dx} = 5 \implies dx = \frac{du}{5}.
\]

### Step 3: Substitute \( u \) into the integral

Substituting \( u \) and \( dx \) into the integral, we get:

\[
y = \int \cos(u) \cdot \frac{du}{5} = \frac{1}{5} \int \cos(u) \, du.
\]

### Step 4: Integrate with respect to \( u \)

The integral of \( \cos(u) \) is:

\[
\int \cos(u) \, du = \sin(u).
\]

Substituting back \( u = 5x \):

\[
y = \frac{1}{5} \sin(5x).
\]

### Step 5: Add the constant of integration

Finally, add the constant of integration, \( C \):

\[
y = \frac{1}{5} \sin(5x) + C.
\]

### Final Answer

\[
\boxed{y = \frac{1}{5} \sin(5x) + C}
\]

### Explanation of what to write in your paper:

1. Write the given differential equation: \( \frac{dy}{dx} = \cos(5x) \).
2. Show the integration step: \( y = \int \cos(5x) \, dx \).
3. Use substitution: Let \( u = 5x \), then \( dx = \frac{du}{5} \).
4. Substitute \( u \) into the integral and simplify: \( y = \frac{1}{5} \int \cos(u) \, du \).
5. Integrate and substitute back: \( y = \frac{1}{5} \sin(5x) + C \).
6. Box the final answer.

Would you like further details or have any questions about the process?

### 5 Related Questions:
1. How do you solve differential equations with different trigonometric functions?
2. What is the substitution method and when is it useful in integration?
3. How do you determine the constant of integration in an indefinite integral?
4. What are the general steps to solve first-order differential equations?
5. How does the chain rule relate to integrating trigonometric functions?

### Tip:
Always remember to add the constant of integration (\( C \)) when solving indefinite integrals, as it represents an infinite number of possible solutions.

Learn how to solve the differential equation dy/dx = cos(5x) step-by-step with integration by substitution. This problem involves differential equations, integration techniques, and trigonometric functions. Suitable for undergraduate level studies.

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