Question

If \(\int \frac{\log(x + \sqrt{1 + x^2})}{1 + x^2} \, dx = f(g(x)) + c\), then:

• A. \(f(x) = \frac{x^2}{2}, \, g(x) = \log(x + \sqrt{1 + x^2})\)
• B. \(f(x) = \log(x + \sqrt{1 + x^2}), \, g(x) = \frac{x^2}{2}\)
• C. \(f(x) = x^2, \, g(x) = \log(x + \sqrt{1 + x^2})\)
• D. \(f(x) = \log(x - \sqrt{1 + x^2}), \, g(x) = x^2\)

Hint:

When dealing with integrals of logarithmic functions, look for substitutions involving the argument of the logarithm and its derivative. This can simplify the problem significantly