What is the integral from 1 to 5 of (x-5/x)^2 ?

The integral from 1 to 5 of (x-5/x)^2 is 64/3

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integral from 1 to 5 of (x-5/x)^2

\int_{1}^{5} (x - \frac{5}{x})^{2} d x

\frac{64}{3}

Solution steps

\int_{1}^{5} (x - \frac{5}{x})^{2} d x

Expand

(x - \frac{5}{x})^{2} : x^{2} - 10 + \frac{25}{x ^{2}}

= \int_{1}^{5} x^{2} - 10 + \frac{25}{x ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{5} x^{2} d x - \int_{1}^{5} 10 d x + \int_{1}^{5} \frac{25}{x ^{2}} d x

\int_{1}^{5} x^{2} d x = \frac{124}{3}

\int_{1}^{5} 10 d x = 40

\int_{1}^{5} \frac{25}{x ^{2}} d x = 20

= \frac{124}{3} - 40 + 20

Simplify

\frac{124}{3} - 40 + 20 : \frac{64}{3}

= \frac{64}{3}

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What is the integral from 1 to 5 of (x-5/x)^2 ?
The integral from 1 to 5 of (x-5/x)^2 is 64/3