Q 9 - Calculus

Question

Finding antiderivatives by undoing the chain rule: For each function f that follows, find a function F with the property that F(x) = f(x). You may have to guess and check to find such a function 

Step-by-Step Solution

Answer

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Step 1: Method overview

To find \(F\) such that \(F'(x) = f(x)\) by undoing the chain rule, look for a composition \(g(h(x))\) whose derivative \(g'(h(x)) \cdot h'(x)\) matches \(f(x)\).

Step 2: Apply the technique

Identify the inner function \(h(x)\) and check if \(h'(x)\) appears as a factor. Then the antiderivative is \(F(x) = g(h(x)) + C\), where \(g\) is an antiderivative of the outer function evaluated at \(h(x)\). This is the reverse of the chain rule, also known as \(u\)-substitution.