Problem 32 - Essential Calculus

Question

Use a computer algebra system to determine the antiderivative that passes through the given point. Use the system to graph the resulting antiderivative. $$ \int x \sqrt{x^{2}+2 x} d x,(0,0) $$

Step-by-Step Solution

Answer

The antiderivative of the given integral is \( \frac{(x^{2} + 2x)^{1.5}}{3} \), and this passes through the point (0,0). The graph of this antiderivative can be created using a computer algebra system.

Step 1: Identify the Integral

The given integral is \( \int x \sqrt{x^{2}+2 x} d x \). The antiderivative of this function, also known as the indefinite integral, will be a function F(x) such that the derivative of F(x) is equal to the integrand function.

Step 2: Apply the Substitution Method

It's useful to apply the substitution method in this case. Let \( u = x^{2}+2 x \). Then, \( du = (2x+2) dx \) and \( dx = du/(2x+2) \). This changes the integral to \( \frac{1}{2} \int u^{0.5} du \).

Step 3: Integral Calculation

Compute the integral now, which becomes \( \frac{1}{2} \int u^{0.5} du = \frac{1}{2} * \frac{2}{3} * u^{0.5 + 1} = \frac{u^{1.5}}{3} \). Substituting \( u = x^{2}+2x \) back into the equation gives \( \frac{(x^{2} + 2x)^{1.5}}{3} \).

Step 4: Include the Constant of Integration

Since we have found the indefinite integral or antiderivative, we need to not forget about the constant of integration. So, the most general antiderivative is \( \frac{(x^{2} + 2x)^{1.5}}{3} + C \).

Step 5: Verify the Antiderivative Passes through the Point

The given point is (0,0). When x = 0, y should also equal 0. Substituting x = 0 into the antiderivative equation gives \( \frac{(0)^{1.5}}{3} + C = 0 \), which implies that C = 0.

Step 6: Graph the Resulting Antiderivative

Having found the antiderivative, graph it using the computer algebra system. The function \( \frac{(x^{2} + 2x)^{1.5}}{3} \) should be graphed over a suitable domain and range.