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Chapter 4: Problem 75

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{9} \frac{1}{\sqrt{x}(1+\sqrt{x})^{2}} d x $$

Short Answer

Expert verified
The solution to the definite integral \(\int_{1}^{9} \frac{1}{\sqrt{x}(1+\sqrt{x})^{2}} d x\) is 0.5.

Step by step solution

01

Simplify the integrand

The first step in this process is simplifying the function that we are integrating, if possible. Here, let's set \(u = 1 + \sqrt{x}\). This will simplify the integrand and make finding the antiderivative easier. When \(u = 1+\sqrt{x}\), we get \(du = \frac{1}{2\sqrt{x}}\, dx\). Multiplying both sides by 2 gives \(2du = \frac{1}{\sqrt{x}}\, dx\). Substitute these into the integral and the expression becomes \(\int_{2}^{4} \frac{2}{u^{2}} du\).
02

Find the antiderivative

Now the integrand is easier to handle, we can find the antiderivative. The integral of \(\frac{2}{u^{2}}\) with respect to \(u\) is \(-2/u\).
03

Use the fundamental theorem of calculus to find the definite integral

The next step in evaluating the definite integral is to plug in the limits of integration. Apply the Fundamental Theorem of Calculus, which says that if \(F\) is an antiderivative of \(f\), then \(\int_{a}^{b} f(x) d x = F(b)-F(a)\). Plugging in the upper limit of integration (4), we get \(-2/4 = -0.5\). Plugging in the lower limit of integration (2), we get -2/2 = -1. So the definite integral becomes \(-0.5 - (-1) = 0.5\).
04

Verify the result using a graphing utility (optional)

We can also verify this result using a graphing utility. This step is optional but is useful for checking the work. The integral represents the area under the curve of the function from 1 to 9. When the function \(\frac{1}{\sqrt{x}(1+\sqrt{x})^{2}}\) is graphed and the area from 1 to 9 is calculated, it should equal 0.5, confirming our result.

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