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

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

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integral from 1 to e of (1+sqrt(x))/x

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

2 e - 1

Decimal Notation

2.29744 \dots

Solution steps

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

Expand

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

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

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

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

= 1 + 2 e - 2

Simplify

= 2 e - 1

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