Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. $$ \int y^{2} \sqrt{y} d y $$

Step-by-Step Solution

Answer

The indefinite integral \( \int y^{2} \sqrt{y} d y \) results in \( \frac{2}{7} y^{3.5} + C \)

Step 1: Rewrite the Expression

Rewrite the given integral in the power format for easier calculation. The given integral \( \int y^{2} \sqrt{y} d y \) can be rewritten as \( \int y^{2} \times y^{1/2} dy \). When multiplying like bases, add the exponents. The rewritten integral then becomes \( \int y^{2.5} dy \).

Step 2: Application of Power Rule

Apply the Power Rule which states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(n\) is any real number and (+C) represents the constant of integration. Here, \(n=2.5\), therefore the power rule gives \( \frac{y^{2.5+1}}{2.5+1} + C \) = \( \frac{y^{3.5}}{3.5} + C \).

Step 3: Simplify the Expression

The expression simplifies to \( \frac{2}{7} y^{3.5} + C \)