Problem 40 - Essential Calculus

Question

Use the tabular method to find the integral. $$ \int x^{2}(x-2)^{3 / 2} d x $$

Step-by-Step Solution

Answer

The integral of \(x^{2}(x-2)^{3 / 2}\)is \(\frac{2}{5} x^{2} (x-2)^{5/2} - \frac{2}{5} x (x-2)^{5/2} +\frac{4}{35} (x-2)^{9/2} + C\) where C is the constant of integration.

Step 1: Identify Function and Derivative

Select a function that gets simpler every time it is derived, and another function that remains manageable every time it is integrated. In this case, select the function \(f(x) = x^{2}\), and \(g(x) = (x-2)^{3 / 2}\). Derive f(x): \[f'(x) = 2x\], \[f''(x) = 2\], \[f'''(x) = 0\]. Integrate g(x): \[∫g(x) dx =\frac{2}{5}(x-2)^{5 / 2}\], \[∫∫g(x) dx =\frac{2}{35}(x-2)^{7 / 2}\].

Step 2: Setup the Table

Arrange these values in a table: \[ \begin{array}{lcl} + & x^{2} & (x - 2)^{3/2}\ - & 2x & \frac{2}{5}(x - 2)^{5/2}\ + & 2 & \frac{2}{35}(x - 2)^{7/2}\ - & 0 & 0 \end{array}\] The signs alternate starting with +, and the result of \(f'''(x)\) results in 0, which gives us a stopping condition.

Step 3: Finding the Integral

Multiply diagonal terms together, keep track of the signs, and add the products to find the integral. Here it is: \[ \int x^{2}(x-2)^{3/2} dx = \frac{2}{5} x^{2} (x-2)^{5/2} - \frac{2}{5} x (x-2)^{5/2} +\frac{4}{35} (x-2)^{9/2}\]