Problem 11 - Essential Calculus

Question

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{1}^{4} \frac{u-2}{\sqrt{u}} d u $$

Step-by-Step Solution

Answer

The result obtained through this process is \(10-2 = 8\)

Step 1: Divide the Function

Start by separating the function into simpler parts that can be integrated individually. In this case, the equation \( \frac{u-2}{\sqrt{u}} \) can be written as: \( \int_{1}^{4} (\frac{u}{\sqrt{u}} - \frac{2}{\sqrt{u}}) du \). which simplifies to \( \int_{1}^{4} (u^{\frac{1}{2}} - 2u^{-\frac{1}{2}}) du \).

Step 2: Apply the Power Rule for Integration

Then apply the power rule for integration on both sub-functions. The power rule states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). Therefore, it becomes \( [2u^{\frac{3}{2}} - 4u^{\frac{1}{2}}]_1^4 \).

Step 3: Compute between the Limits of Integration

Evaluate the expression above between 4 and 1. Therefore, it is \( [2*4^{\frac{3}{2}} - 4*4^{\frac{1}{2}}] - [2*1^{\frac{3}{2}} - 4*1^{\frac{1}{2}}] \). Simplify it to get the final answer.