Question

Find the integral \( \int 2 \, dy = (y + \cos x) \, dx \)

• A. \( y = \sin x + C \)
• B. \( y = \cos x + C \)
• C. \( y = x + C \)
• D. \( y = \sin x + \cos x + C \)

Hint:

When dealing with integration involving multiple variables, break down the integral and apply the basic integration formulas for each term.

Answer 1

The Correct Option is D

Step 1: Understanding the integral.
We are given the equation: \[ \int 2 \, dy = (y + \cos x) \, dx \] Integrating both sides with respect to their respective variables: \[ 2y = \int (y + \cos x) \, dx \]
Step 2: Integrate the right-hand side.
The integral of \( y \) with respect to \( x \) is \( yx \), and the integral of \( \cos x \) with respect to \( x \) is \( \sin x \). Therefore: \[ 2y = yx + \sin x + C \]
Step 3: Solve for \( y \).
Thus, we have: \[ y = \sin x + \cos x + C \]
Final Answer: \( y = \sin x + \cos x + C \).