Definite Integration
Integration
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: The derivative of ln(1+(g(x))^2) is (1/(1+(g(x))^2)) * 2g(x) * g'(x). Since g(x) is an antiderivative of f(x), g'(x) = f(x). Substituting this gives (2g(x)f(x))/(1+(g(x))^2).
Let F(x) = ln(1+(g(x))^2). Then F'(x) = (1/(1+(g(x))^2)) * d/dx(1+(g(x))^2) = (1/(1+(g(x))^2)) * 2g(x) * g'(x). Since g(x) is an antiderivative of f(x), g'(x) = f(x). Thus, F'(x) = (2g(x)f(x))/(1+(g(x))^2).
Correct Answer: B