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

Step 1: Understanding the Concept:
This problem uses the method of substitution in integration. By replacing a portion of the integrand with a new variable, we can transform the integral into a simpler known form.
Step 3: Detailed Explanation:
1. Let \(I = \int \cos x f(\sin x) dx\).
2. Use substitution: Let \(u = \sin x\).
3. Differentiate both sides: \(du = \cos x dx\).
4. Substitute these into the integral:
\[ I = \int f(u) du \]
5. We are given that \(\int f(x) dx = g(x) + C\). Therefore, \(\int f(u) du = g(u) + C\).
6. Substitute the original variable back in:
\[ I = g(\sin x) + C \]
Step 4: Final Answer:
The result of the integral is \(g(\sin x) + C\).