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

The integral from 1 to e of (x^2+x+1)/x is e+(e^2-1)/2

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

\int_{1}^{e} \frac{x ^{2} + x + 1}{x} d x

e + \frac{e ^{2} - 1}{2}

Solution steps

\int_{1}^{e} \frac{x ^{2} + x + 1}{x} d x

Expand

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

= \int_{1}^{e} x + 1 + \frac{1}{x} 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}^{e} x d x + \int_{1}^{e} 1 d x + \int_{1}^{e} \frac{1}{x} d x

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

\int_{1}^{e} 1 d x = e - 1

\int_{1}^{e} \frac{1}{x} d x = 1

= \frac{e ^{2} - 1}{2} + e - 1 + 1

Simplify

= e + \frac{e ^{2} - 1}{2}

\int_{- 2}^{3} \frac{19}{x ^{4}} d x

x \to 2 lim (- \frac{1}{x - 2})

\int (3 - x^{2})^{3} d x

f (x) = e^{x^{3}}

x \to - 5 + lim (\frac{- 2}{x ^{2} - 25})

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