Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. $$ \int \sqrt{x}(x+1) d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\sqrt{x}(x+1) dx\) is \(\frac{x^3}{3} + x + C\)

Step 1: Expand the function

Expand the function inside the integral, \(\sqrt{x}(x+1) = x\sqrt{x} + \sqrt{x}\). So our integral becomes \(\int (x\sqrt{x} + \sqrt{x}) dx\).

Step 2: Separate the Integral

Based on the properties of integrals, we can separate this integral into two parts: \(\int x\sqrt{x} dx + \int \sqrt{x} dx\)

Step 3: Change of Variable

Take \(u = x^{3/2}\). Then, \(\frac{du}{dx} = \frac{3}{2} x^{1/2}\) or \(dx = \frac{2du}{3x^{1/2}}\). Substitute back to the first integral, we have \(\int \frac{2}{3}u du\). Take \(v = x^{1/2}\). Then, \( \frac{dv}{dx} = \frac{1}{2} x^{-1/2}\) or \(dx = 2 dv x^{1/2}\). Substitute back to the second integral, we have \(\int 2v du \).

Step 4: Integrate

Now we can integrate the functions. For the first integral, the integral of \(u\) is \(\frac{u^2}{2} = \frac{x^3}{2}\). For the second integral, the integral of \(v\) is \(\frac{v^2}{2} = \frac{x}{2}\). So, the indefinite integral will be: \(\frac{2}{3} \cdot \frac{x^3}{2} + 2 \cdot \frac{x}{2} + C\)