Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{5} \frac{x}{\sqrt{2 x-1}} d x $$

Short answer

The evaluated definite integral is \(4\).

Step-by-step solution

Identify Substitution Variable

First, a good candidate for substitution must be identified. In this case, we can select \(u = 2x - 1\). Note that the derivative of \(u\), \(du\), is \(2dx\).

Adjust the Differential

Next, solve for \(dx\) in terms of \(du\) and \(x\), so we get \(dx = \frac{1}{2}du\). This substitution adjusts the integral so we can handle the square root.

Change the Limits of Integration

When we change the variable of integration from \(x\) to \(u\), the limits of integration will change as well. Using \(u = 2x - 1\), evaluate the limits; when \(x = 1\), \(u = 1\), and when \(x = 5\), \(u = 9\). The new limits of integration are from \(1\) to \(9\).

Substitute Variable and Limits in the Integral

After substitution, the new integral is \(\int_{1}^{9} \frac{1}{2}u^{-\frac{1}{2}} du\).

Integrate

Now, solve the integral. We first add 1 to the exponent to get \(u^{1/2}\). Then flip that exponent to get \(2u^{1/2}\). Finally, apply the limits of integration from 1 to 9. The result is \(2(\sqrt{9}-\sqrt{1}) = 2(3 - 1) = 4\).

Verification with a Graphing Utility

Use a mathematical graphing tool such as Desmos or Geogebra, plot the function \(f(x) = \frac{x}{\sqrt{2x - 1}}\) and calculate the definite integral from 1 to 5. The graphing software should confirm the result as \(4\).