2. The first thing to do in order to solve this problem, is to solve for the derivative of the function… ln (2x 2 + 3)
3. This function is a composite function. It means, it has an outer function, ln(x) and an inner function, 2x 2 + 3 To solve this, we need to use the Chain Rule… dy f(g(x)) = f ’(g(x)) g ’(x) dx
4. f ‘ (x) = ln (2x 2 + 3) = 1 . 4x 2x 2 + 3 = 4x 2x 2 + 3 -Parent Function -According to the natural log rule, the derivative of ln (x) is 1/x. The derivative of the inner function is 4x according to the power rule of differentiation. -Simplifying the derivative formula
5. Now that we have the derivative function, we can find the area under the curve. Since we’re not allowed to use a graphing calculator, we need to find this algebraically…
6. Since we already have the antiderivative of the formula, we can just simply use the find the integral from 0 to 4… A = 4x 2x 2 + 3 = ln (2x 2 + 3) = ln (2(4) 2 + 3) - ln (2(0) 2 + 3 = 3.5553 - 1.0986 = 2.4567 0 4 Integral Notation Antidifferentiation Substitution Simplify Area Under the Curve
