Question

If \( f(x) \) is the antiderivative of \( g(x) \) and \[ \int f(x) g(x)(1 + f^2(x)) dx = F(x) \], then \( F(x) = \)

• A. \( \frac{(1 + f^2(x))^2}{4} + C \)
• B. \( \frac{(1 + f^2(x))^2}{2} + C \)
• C. \( \frac{f^2(x) g(x)}{4} + C \)
• D. \( \frac{g^2(x) f(x)}{4} + C \)

Hint:

Integration Using Substitution}
Recognize patterns like \( f(x)g(x)(1 + f^2(x)) \) and attempt substitution
Let \( u = 1 + f^2(x) \) and look for \( f(x)g(x) \) as derivative
Chain rule in reverse (u-substitution) simplifies integrals like this

Answer 1

The Correct Option is A

Let: - \( f(x) = \int g(x) dx \) (so \( f'(x) = g(x) \)) Now consider: \[ F(x) = \int f(x)g(x)(1 + f^2(x)) dx \Rightarrow \text{Let } u = 1 + f^2(x) \Rightarrow \frac{du}{dx} = 2f(x)g(x) \] Then: \[ f(x)g(x)(1 + f^2(x)) = \frac{1}{2} \cdot u \cdot \frac{du}{dx} \Rightarrow \int f(x)g(x)(1 + f^2(x)) dx = \frac{1}{4}(1 + f^2(x))^2 + C \]