Question

If \( \int f(x)\,dx = g(x) \), then \( \int \cos x \, f(\sin x)\,dx \) is equal to:

• A. \( g(\cos x) + C \)
• B. \( g(\sin x) + C \)
• C. \( \int g(x) + \sin x + C \)
• D. \( f(\sin x) + g(\cos x) + C \)

Hint:

Pattern: \(f(\sin x)\cos x\,dx \Rightarrow \text{put } t = \sin x\).

Answer 1

The Correct Option is B

Concept: Use substitution method: \[ \int f(x)\,dx = g(x) \Rightarrow \int f(t)\,dt = g(t) \]

Step 1: Substitution \[ t = \sin x \Rightarrow dt = \cos x \, dx \]

Step 2: Transform integral \[ \int \cos x \, f(\sin x)\,dx = \int f(t)\,dt \]

Step 3: Integrate \[ = g(t) + C = g(\sin x) + C \] Final: \[ {g(\sin x) + C} \]