Q 8 - 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: General approach

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

Step 2: Method

Identify the inner function \(u = h(x)\) and check if \(f(x)\) can be written as \(g'(u) \cdot h'(x)\). Then \(F(x) = g(h(x)) + C\).