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Chapter 7: Problem 24

Evaluate the following integrals. $$\int_{4}^{9} \frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int_{4}^{9} \frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} d x$$ Answer: $$\int_{4}^{9} \frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} d x = \frac{65}{2} + \ln \left(\frac{4}{9}\right)$$

Step by step solution

01

Simplify the integrand

Divide each term in the numerator by \(x^{3/2}\): $$\frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} = x^{5/2-3/2} - x^{1/2-3/2} = x - x^{-1}$$
02

Find the antiderivative of the simplified expression

To find the antiderivative, first we will find the antiderivative of each term in the expression. The antiderivative of \(x\) is \(\frac{x^2}{2}\) and the antiderivative of \(x^{-1}\) is \(\ln |x|\). So, the antiderivative of \(x - x^{-1}\) is \(\frac{x^2}{2} - \ln |x| + C\) where C is the constant of integration.
03

Apply the Fundamental Theorem of Calculus

Now, we will apply the Fundamental Theorem of Calculus by plugging in the given limits 9 and 4. $$\int_{4}^{9} (x - x^{-1}) d x = \left[\frac{x^2}{2} - \ln |x|\right]_{4}^{9}$$
04

Evaluate the integral

Evaluate the expression at the upper limit and then subtract the expression evaluated at the lower limit: $$\left[\frac{9^2}{2} - \ln |9|\right] - \left[\frac{4^2}{2} - \ln |4|\right] = \left[\frac{81}{2} - \ln (9)\right] - \left[\frac{16}{2} - \ln (4)\right]$$ Now, simplify the expression: $$\left[\frac{81}{2} - \ln (9)\right] - \left[\frac{16}{2} - \ln (4)\right] = \frac{81}{2} - \ln (9) -\frac{16}{2} + \ln (4) = \frac{65}{2} + \ln \left(\frac{4}{9}\right)$$ So, the value of the integral is: $$\int_{4}^{9} \frac{x^{5 / 2}-x^{1 / 2}}{x^{3 / 2}} d x = \frac{65}{2} + \ln \left(\frac{4}{9}\right)$$

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