Definite Integration
Antiderivative
Grade Class 12
Question:
Let g(x) be an antiderivative for f(x). Then ln(1+(g(x))^2) is an antiderivative for
(A) \frac{2f(x)g(x)}{1+(f(x))^2}
(B) \frac{2f(x)g(x)}{1+(g(x))^2}
(C) \frac{2f(x)}{1+(f(x))^2}
(D) none
Step-by-Step Solution
Key Concept: Differentiate the given function ln(1+(g(x))^2) with respect to x using the chain rule, knowing that g'(x) = f(x).
Let y = ln(1+(g(x))^2). Then dy/dx = (1/(1+(g(x))^2)) * 2g(x) * g'(x). Since g'(x) = f(x), dy/dx = 2f(x)g(x)/(1+(g(x))^2).
Correct Answer: B