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

The integral from 1 to 2 of (x+2/x)^2 is 25/3

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integral from 1 to 2 of (x+2/x)^2

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

\frac{25}{3}

Decimal

8.33333 \dots

Solution steps

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

Expand

(x + \frac{2}{x})^{2} : x^{2} + 4 + \frac{4}{x ^{2}}

= \int_{1}^{2} x^{2} + 4 + \frac{4}{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}^{2} x^{2} d x + \int_{1}^{2} 4 d x + \int_{1}^{2} \frac{4}{x ^{2}} d x

\int_{1}^{2} x^{2} d x = \frac{7}{3}

\int_{1}^{2} 4 d x = 4

\int_{1}^{2} \frac{4}{x ^{2}} d x = 2

= \frac{7}{3} + 4 + 2

Simplify

\frac{7}{3} + 4 + 2 : \frac{25}{3}

= \frac{25}{3}

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What is the integral from 1 to 2 of (x+2/x)^2 ?
The integral from 1 to 2 of (x+2/x)^2 is 25/3