Find ∫(sin x + 2)dx

Analyze the question: We need to integrate a sum of two terms, sin x and the constant 2, with respect to x.

Concept: Integrate term by term:

• ∫sin x dx = -cos x + C (because the derivative of -cos x is sin x)
• ∫k dx = kx + C for any constant k

Solution:

∫(sin x + 2) dx = ∫sin x dx + ∫2 dx

= -cos x + 2x + K

Check by differentiating: d/dx(-cos x + 2x + K) = sin x + 2 ✓

Checking the options:

• A. cos x + x² + K: wrong sign on cos x, and x² is incorrect (that would come from integrating 2x).
• B. cos x 2x + K: not a valid expression (it is probably a typo), and it lacks the negative sign.
• C. -cos x + x² + K: the cos part is right, but ∫2 dx is 2x, not x².
• D. -cos x + 2x + K: matches our result exactly.

The correct answer is D.

Tip: Remember the sign pattern: ∫sin x dx = -cos x, but ∫cos x dx = +sin x. Also, integrating a constant just attaches x to it, so ∫2 dx = 2x. Always differentiate your answer to confirm it quickly in the exam.