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

The integral from 1/2 to 3 of (x^2-1)/(x^2) is 5/6

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integral from 1/2 to 3 of (x^2-1)/(x^2)

\int_{\frac{1}{2}}^{3} \frac{x ^{2} - 1}{x ^{2}} d x

\frac{5}{6}

Solution steps

\int_{\frac{1}{2}}^{3} \frac{x ^{2} - 1}{x ^{2}} d x

Expand

\frac{x ^{2} - 1}{x ^{2}} : 1 - \frac{1}{x ^{2}}

= \int_{\frac{1}{2}}^{3} 1 - \frac{1}{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_{\frac{1}{2}}^{3} 1 d x - \int_{\frac{1}{2}}^{3} \frac{1}{x ^{2}} d x

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

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

= \frac{5}{2} - \frac{5}{3}

Simplify

\frac{5}{2} - \frac{5}{3} : \frac{5}{6}

= \frac{5}{6}

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What is the integral from 1/2 to 3 of (x^2-1)/(x^2) ?
The integral from 1/2 to 3 of (x^2-1)/(x^2) is 5/6