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

The integral from 1 to 3 of (1+1/x+1/(x^2)) is ln(3)+8/3

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

\int_{1}^{3} (1 + \frac{1}{x} + \frac{1}{x ^{2}}) d x

ln (3) + \frac{8}{3}

Decimal

3.76527 \dots

Solution steps

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

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

\int_{1}^{3} \frac{1}{x} d x = ln (3)

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

= 2 + ln (3) + \frac{2}{3}

Combine the fractions

2 + \frac{2}{3} : \frac{8}{3}

= ln (3) + \frac{8}{3}

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What is the integral from 1 to 3 of (1+1/x+1/(x^2)) ?
The integral from 1 to 3 of (1+1/x+1/(x^2)) is ln(3)+8/3